Algebra See if you can notice the difference between most of the math problems we’ve considered so far, and these: Eek! There are letters in there! As you no doubt know, the letters are called variables, and they make possible the wonderful world of algebra. Whoever thought of adding variables into math certainly made tens of millions of school kids’ lives harder, because without algebra, there would be no geometry, no trigonometry, no calculus, and possibly no misery at all in the world. Thanks a lot, Mr. or Mrs. Math Genius . . . Of course, there would probably be no microwave ovens or iPods either, so it’s not all bad. Since the variables represent unspecified quantities, algebra brings arithmetic into the world of the unknown. Of course, much of algebra deals with making that unknown known; that is, solving equations so that variables can be replaced by good old-fashioned numbers. We’ll get to all that soon enough, but first a little vocabulary is in order. Algebra Terms There are six main terms that describe the world of algebra. The GRE won’t ask you to define them but will nonetheless give you questions that require you to work with them. 1. Constant. A numerical quantity that does not change. 2. Variable. An unknown quantity written as a letter. A variable can be represented by any letter in the English alphabet; x or y are common on the GRE, but you’ll see others as well. Variables may be associated with specific things, like x number of apples or y dollars. Other times, variables have no specific association, but you’ll need to manipulate them to show that you understand basic algebraic principles. 3. Coefficient. A coefficient is a number that appears next to a variable and tells how many of that variable there are. For example, in the term 4x, 4 is the coefficient and tells us there are four xs. In the term 3x3, 3 is the coefficient and tells us there are three x3s. 4. Term. The product of a constant and a variable. Or, a quantity separated from other quantities by addition or subtraction. For example, in the equation 3x3 + 2x2 – 7x + 4 = x – 1, the side to the left of the equal sign contains four terms (3x3, 2x2, –7x, 4), while the right side contains two terms (x, –1). The constants, 4 and –1, are considered terms because they are coefficients of variables raised to the zero power: 4 = 4x0. So technically, every algebraic term is the product of a constant and a variable raised to some power. 5. Expression. A funny look on your face. Or in math, any combination of terms. An expression can be as simple as a single constant term, like 5, or as complicated as the sum or difference of many terms, each of which is a combination of constants and variables, such as . Expressions don’t include an equal sign—this is what differentiates expressions from equations. If you’re given the value of every variable in the expression, then you can calculate the numerical value of the expression. Lacking those values, expressions can’t be solved, although they can often be simplified. 6. Equation. Two expressions linked by an equal sign. Much of the algebra on the GRE consists of solving equations. Inputs and Outputs: Simple Substitutions We mentioned above that we can calculate the numerical value of an expression if we’re given the value of every variable in it. In this case, the expression itself is like a machine that takes an input (the variable value) and outputs a solution. One of the simplest kinds of algebraic problems on the GRE operates like this, as in the following example: We mentioned above that we can calculate the numerical value of an expression if we’re given the value of every variable in it. In this case, the expression itself is like a machine that takes an input (the variable value) and outputs a solution. One of the simplest kinds of algebraic problems on the GRE operates like this, as in the following example: If x = 2, what is the value of ? There’s nothing to do but simply input 2 into the expression in place of x and do the math: So here, an input of 2 yields an output of 6. You may see something like this in the beginning of the Math section, but things will most likely get more complicated after that. For example, the inputs themselves may be a bit more complex than a single number; in fact, an input may itself contain variables: If 2y + 8x = 11, what is the value of 3(2y + 8x)? You might see this equation bubbling over with variables and panic. Don’t. Since the expression 2y + 8x appears in both parts of the question, and we’re told this expression equals 11, we can simply substitute that figure in place of the expression on the right to get 3(11) = 33. A question may also involve multiple substitutions. For instance: z = , y = 3x, and x = 2, then what is the value of z? To approach this problem, you just have to input 2 for x to find y, and then input those values into the equation for z. Substituting 2 for x into y = 3x gives y = 3(2) = 6. Inputting x= 2 and y = 6 into the equation for z gives: Simplifying Algebraic Expressions Before we move on to solving more complicated equations, we need to cover a few simplification tools that allow us to change algebraic expressions into simpler but equivalent forms. Distributing The rule of distribution states: The a in this expression can be any kind of term, meaning it could be a variable, a constant, or a combination of the two. When you distribute a factor into an expression within parentheses, multiply each term inside the parentheses by the factor outside the parentheses. 4(x + 2), for example, would become 4x + (4)(2), or 4x + 8. Let’s try a harder one: 3y(y2 – 6). Distributing the 3y term across the terms in the parentheses yields: Seems logical enough. But the true value of distributing becomes clear when you see a distributable expression in an equation. We’ll see an example of this later in the section on linear equations. Factoring Factoring an expression is the opposite of distributing. 4x3 – 8x2 is one mean-looking expression, right? Or so it seems, until you realize that both terms share the greatest common factor 4x2, which you can factor out: By distributing and factoring, you can group or ungroup quantities in an equation to make your calculations simpler, depending on what the other terms in the equation look like. Sometimes distributing will help; other times, factoring will be the way to go. Here are a few more examples of both techniques: Combining Like Terms After factoring and distributing, you can take additional steps to simplify expressions or equations. Combining like terms is one of the simplest techniques you can use. It involves adding or subtracting the coefficients of variables that are raised to the same power. For example, by combining like terms, the expression can be simplified by adding the coefficients of the variable x3 (–1 and 3) together and the coefficients of x2 (1 and 4) together to get: Variables that have different exponential values are not like terms and can’t be combined. Two terms that do not share a variable are also not like terms and cannot be combined regardless of their exponential value. For example, you can’t combine: You can, however, factor the first expression to get x2(x2 + 1), which you should do if it helps you answer the question. Linear Equations with One Variable Simplifying is nice, and helpful to boot, but solving is really where it’s at. To solve an equation, you have to isolate the variable you’re solving for. That is, you have to “manipulate” the equation until you get the variable alone on one side of the equal sign. By definition, the variable is then equal to everything on the other side of the equal sign. You can’t manipulate an equation the way you used to manipulate your little brother or sister. When manipulating equations, there are rules. Here’s the first and most fundamental. In fact, it’s so important we’re going to bold it: Whatever you do to one side of an equation, you must do to the other side. If you divide one side of an equation by 3, divide the other side by 3. If you take the square root of one side of an equation, take the square root of the other. If you fall in love with one side of the equation, fall in love with the other. Neither side will think you’re a two-timer. They’ll think you’re a highly skilled mathematician. By treating the two sides of the equation in the same way, you don’t change what the equation means. You change the form of the equation into something easier to work with—that’s the point of manipulating it—but the equation remains true since both sides stay equal. Take, for instance, the equation 3x + 2 = 5. You can do anything you want to it as long as you do the same thing to both sides. Here, since we’re trying to get the variable x alone on the left, the thing to do is subtract 2 from that side of the equation. But we can only do that if we subtract 2 from the other side as well: Ah, that’s better. Now we can just divide both sides by 3 to get x = 1, and we’re done. You should use the simplification techniques you learned above (distributing, factoring, and combining like terms) to help you solve equations. For example: That seems fairly nasty, since there aren’t any like terms to combine. But wait a sec . . . what if you distribute that 3y on the left side of the equation? That would give: Shiver our timbers! Now we can subtract 3y3 from both sides to get and then simply divide both sides by 18 to get y = 2. Reverse PEMDAS Many equations include a combination of elements you learned about in our arithmetic discussion. Remember PEMDAS, the acronym you learned to help you remember the order of operations? Well, what do you get if you do PEMDAS in reverse? SADMEP, of course. Or, using the old corny mnemonic device, Sally Aunt Dear My Excuse Please. Wait, scratch that—mnemonics don’t work in reverse. Why do we want to reverse our trusty order of operations, anyway? The idea is to undo everything that has been done to the variable so that it will be isolated in the end. So you should first subtract or add any extra terms on the same side as the variable. Then divide and multiply anything on the same side as the variable. Next, raise both sides of the equation to a power or take their roots according to any exponent attached to the variable. Finally, work out anything inside parentheses. In other words, do the order of operations backward: SADMEP! We’ll need to demonstrate with an example. Here’s a little monstrosity that at first glance might make you reconsider your decision to go to grad school: At second glance you might feel the same way. But at third glance you’d know what to do. In this equation, poor little x is being square rooted, multiplied by 2, added to 3, and encased in parentheses—all in the numerator of a fraction. That’s hardly what we’d call “alone time.” You’ve got to get him out of there! Undo all of these operations to liberate x and solve the equation. Let SADMEP be your guide: First, subtract 2 from both sides of the equation: There’s no addition or division possible at this point, but we can multiply both sides by 2 to get rid of the fraction: Now divide both sides by 3 (see you later, parentheses!): Now we’re in position to subtract 3 from each side: Divide both sides by 2: Finally, square each side to get rid of the square root: Success! You’ve freed poor x from all of those bullying operations. Variables in the Denominator Remember, the key to solving equations is to isolate the variable, but how to do this depends on where the variable is located. A variable in the numerator of a fraction is usually pretty easy to isolate. But if the variable is in the denominator, things get more complicated. See what you can make of this one: Following SADMEP, start by subtracting the 3: But now you have to get the x out of the denominator, and the only way to do that is to multiply both sides of the equation by that denominator, x + 2: Divide both sides by 4: Subtract 2 from each side: Equations with Absolute Value To solve an equation in which the variable is within absolute value bars, you have to follow a two-step process: 1. Isolate the expression within the absolute value bars. 2. Divide the equation in two. Divide the equation in two? What is this, a magic trick? Kind of. Watch: If |x + 3| = 5, then x = Since both 5 and –5 within absolute value bars equal 5, the expression inside the bars can equal 5 or –5 and the equation will work out. That’s why we have to work through both scenarios. So we’re actually dealing with two equations: x + 3 = 5 x + 3 = –5 For a complete solution, we need to solve both. In the first equation, x = 2. In the second equation, x = –8. So the solutions to the equation |x + 3| = 5 are x = {–8, 2}. Both work. Substitute them back into the equation if you have any doubts and to reinforce why we need to solve two equations to get a full answer to the question. Equations with Exponents and Radicals Absolute value equations aren’t the only ones with more than one possible answer. Exponents and radicals can also have devilish effects on algebraic equations that are similar to those caused by absolute value. Consider the equation x2 = 25. Seems pretty simple, right? Just take the square root of both sides, and you end up with x = 5. But remember the rule of multiplying negative numbers? When two negative numbers are multiplied together the result is positive. In other words, –5 squared also results in 25: –5 ×–5 = 25. This means that whenever you have to take the square root to simplify a variable brought to the second power, the result will be two solutions, one positive and one negative: . (The only exception is if x = 0.) You’ll see what we mean by working through this question: If 2x2 = 72, what is the value of x? To solve this problem, we first divide both sides by 2 to get x2 = 36. Now we need to take the square root of both sides: . Recognizing when algebraic equations can have more than one possible answer is especially important in Quantitative Comparison questions. Sometimes it may appear that the relationship between columns A and B leans in a definite direction, until you notice that one of the columns can actually have more than one possible value. Keep your eyes peeled for this common nuance. Linear Equations with Two or More Variables So you’re kicking butt and taking names on those old one-variable equations, huh? Good. But some GRE questions contain two variables. Lucky for you, those questions also contain two equations, and you can use the two equations in conjunction to solve for the variables. These two equations together are called a system of equations or simultaneous equations. We said earlier that manipulating equations isn’t like manipulating your younger brother or sister. Actually, that’s not entirely true: Solving simultaneous equations is like manipulating your younger brother and sister. You use one equation against the other, and in the end you get whatever you want. There are two ways to solve simultaneous equations. The first method involves substitution, and the second involves adding or subtracting one equation from the other. Let’s look at both techniques. Solving by Substitution You’ve already seen examples of substitution in the Input/Output discussion earlier in the chapter, so you should be familiar with the mechanics of plugging values or variables from one place into another place to find what you’re looking for. In these next examples, both pieces of the puzzle are equations. Our method will be to find the value of one variable in one equation and then plug that into a second equation to solve for a different variable. Here’s an example: If x – 4 = y – 3 and 2y = 6, what is x? You’ve got two equations, and you have to find x. The first equation contains both x and y. The second equation contains only y. To solve for x, you first have to solve for y in the second equation and substitute that value into the first equation. If 2y = 6, then dividing both sides by 2, y = 3. Now, substitute 3 for y in the equation x – 4 = y – 3: That’s all there is to it. But here’s one that’s more likely to give you trouble: If 3x = y + 5 and 2y – 2 = 12k, what is x in terms of k? Notice anything interesting? There are three variables in this one. To solve for x in terms of k, we have to first get x and k into the same equation. To make this happen, we can solve for y in terms of k in the second equation and then substitute that value into the first equation to solve for x: Then substitute 6k + 1 for y in the equation 3x = y + 5: This is our answer, since x is now expressed in terms of k. Note that you could also solve this problem by solving for y in terms of x in the first equation and substituting that expression in for y in the second equation. Either way works. Solving by Adding or Subtracting The amazing thing about simultaneous equations is that you can actually add or subtract the entire equations from each other. Here’s an example: If 6x + 2y = 11 and 5x + y = 10, what is x + y? Look what happens if we subtract the second equation from the first: To add or subtract simultaneous equations, you need to know what variable or expression you want to solve for, and then add or subtract accordingly. We made the example above purposely easy to show how the method works. But you won’t always be given two equations that you can immediately add or subtract from each other to isolate the exact variable or expression you seek, as evidenced by this next example: If 2x + 3y = –6 and –4x + 16y = 13, what is the value of y? We’re asked to solve for y, which means we’ve got to get rid of x. But one equation has 2x and the other has –4x, which means the x terms won’t disappear by simply adding or subtracting them. Don’t despair; our Golden Rule of Algebra comes to the rescue: If you do the same thing to both sides of an equation, you don’t change the meaning of the equation. That means that in this case, we could multiply both sides of 2x + 3y = –6 by 2, which would give us 2(2x +3y) = 2(–6). Using the trusty distributive law, and multiplying out the second part, gives us 4x +6y = –12. Now we’re in a position to get rid of those pesky x terms by adding this new (but equivalent) form of equation 1 to equation 2: On the GRE, you will almost always be able to manipulate one of the two equations in a pair of simultaneous equations so that they can be added and subtracted to isolate the variable or expression you want. If you can’t see how to do this, or for questions with easy numbers, go ahead and solve by using substitution instead. As you practice with these types of problems, you’ll get a sense for which method works best for you. Binomial and Quadratic Equations A binomial is an expression containing two terms. The terms (x + 5) and (x – 6) are both binomials. A quadratic expression takes the form ax2 + bx + c, where . Quadratics closely resemble the products formed when binomials are multiplied. Coincidence? Fat chance. That’s why we treat these topics together. We’ll start off with binomials, and work our way to quadratics. Multiplying Binomials The best acronym ever invented (other than SCUBA: “self-contained underwater breathing apparatus”) will help you remember how to multiply binomials. This acronym is FOIL, and it stands for First + Outer + Inner + Last. The acronym describes the order in which we multiply the terms of two binomials to get the correct product. For example, let’s say you were kidnapped by wretched fork-tongued lizard-men whose only weakness was binomials. Now what if the lizard-king asked you to multiply these binomials: (x + 1)(x + 3) What would you do? Follow FOIL, of course. First, multiply the first (F) terms of each binomial: x × x = x2 Next, multiply the outer (O) terms of the binomials: x × 3 = 3x Then, multiply the inner (I) terms: 1 × x = x And then multiply the last (L) terms: 1 × 3 = 3 Add all these terms together: Finally, combine like terms, and you get: Here are a few more examples of multiplied binomials to test your FOILing faculties: Note that the last one doesn’t form a quadratic equation, and that none of the terms can be combined. That’s okay. When presented with binomials, follow FOIL wherever it leads. Working with Quadratic Equations A quadratic equation will always have a variable raised to the power of 2, like this: x2 = 10x – 25 Your job will be to solve for the given variable, as we’ve done with other algebraic equations throughout this section. The basic approach, however, is significantly different from what you’ve done so far. Instead of isolating the variable on the left, you’ll want to get everything on the left side of the quadratic equation, leaving 0 on the right. In the example above, that means moving the 10x and –25 to the left side of the equation: x2 – 10x + 25 = 0 Now it’s looking like most of the products of binomials we saw in the previous section, except instead of being just a quadratic expression, it’s a quadratic equation because it’s set equal to 0. To solve the equation, we need to factor it. Factoring a quadratic equation means rewriting it as a product of two terms in parentheses, like this: x2 – 10x + 25 = (x – 5)(x – 5) How did we know to factor the equation into these binomials? Here’s the secret: Factoring quadratic equations on the GRE always fits the following pattern: (x ± m)(x ± n) Essentially, we perform FOIL in reverse. When we approach a quadratic like x2 – 10x + 25, the two numbers we’re looking for as our m and n terms need to multiply to give the last number in the equation. The last number in the equation is 25, so we need to find two numbers whose product is 25. Some pairs of numbers that work are: 1 and 25 –1 and –25 5 and 5 –5 and –5 Further, the sum of the two numbers needs to give the middle number in the equation. Be very careful that you don’t ignore the sign of the middle number: x2 – 10x + 25 Since you’re subtracting 10x, the middle number is –10. That means the m and n numbers we seek not only need to multiply to 25 but also need to add to –10. Going back to our factor list above, –5 and –5 is the only pair that works. Substituting this into the pattern gives: (x – 5)(x – 5) And since we originally set the equation equal to 0, we now have: (x – 5)(x – 5) = 0 For the product of two terms to equal 0, that means that either one could be 0. Here both terms are (x – 5), so x – 5 must equal 0. x – 5 = 0 x = 5 The final answer is x = 5. In this example, m and n are equal, which is why we end up with only one answer. But that’s usually not the case. Let’s look at another example, using different numbers: x2 = –10x – 21 To solve for x, first move everything to the left to set the equation equal to 0: x2 + 10x + 21 = 0 Now we need to figure out what numbers fit our pattern: (x ± m)(x ± n) We know from our equation that m × n needs to equal 21, and m + n needs to equal 10. So, which numbers work? Let’s look at m × n = 21 first: 1 and 21 –1 and –21 3 and 7 –3 and –7 We can eliminate (1 and 21) and (–1 and –21), since neither of these pairs add up to 10. The third pair, 3 and 7, adds to 10, so we can stop right there and plug these numbers into our pattern: (x + 3)(x + 7) = 0 If you need to double-check your factoring, just FOIL the resulting binomials, which should bring you right back to the original quadratic. Since we now have the product of two different binomials sets equal to 0, one of the two terms needs to be 0. So, either (x + 3) = 0 or (x + 7) = 0, which means x could be equal to –3 or –7. There’s no way to determine for sure, since both values work. Quadratic Factoring Patterns There are three patterns of quadratics that commonly appear on the GRE. Learn them now, and you’ll work faster on test day. Pattern 1: x2 + 2xy + y2 = (x + y)(x + y) = (x + y)2 Example: x2 + 6xy + 9 = (x + 3)2 Pattern 2: x2 – 2 xy + y 2 = (x – y )( x – y) = (x – y)2 Example: x2 – 10xy + 25 = (x – 5)2 Pattern 3: (x + y)(x – y) = x2 – y2 Example: (x + 4)(x – 4) = x2 – 16 You may be wondering why we wrote the last pattern with the factored form first. We wrote it this way because this is the way it often appears on the GRE: You’ll be given an expression that fits the pattern on the left of the equal sign [(x + y)(x – y)], and you’ll need to recognize its equivalent form of (x2 – y2). Here’s an example of how you can use this pattern on test day: What is the value of ? This looks horrific, and, well, it is if you attempt to perform a full FOIL treatment on it. Who wants to multiply 72 by 85 and deal with all those radical signs? Not us. Not you. Not anyone. Luckily, quadratic pattern 3 helps us avoid all that work. That pattern states that whenever we have the sum of two values multiplied by the difference of those same values, the whole messy expression is equal to x2 – y2. Here, if we let x = and y = , then x2 – y2 = . If this doesn’t look any better to you, then you’re not realizing that the squared symbol (the little 2) and the square root symbol (the ) cancel each other out. This gives the much simpler expression 85 – 72, which equals 13. This is an excellent example of how changing your math mindset, something we implored you to do in the previous chapter, will help you on the GRE. In the difficult-looking problem we just tackled, your first instinct may have been to hack your way through the numbers. However, if you instead suspected that the GRE test makers probably wouldn’t present a problem like this if there wasn’t a more elegant solution, then you might have searched for an easy way in. Quadratic factoring pattern 3 does the trick. Inequalities Life isn’t always fair. That’s why there are inequalities. An inequality is like an equation, but instead of relating equal quantities, it specifies exactly how two quantities are not equal. There are four types of inequalities: 1. x > y x is greater than y. 2. x < y x is less than y. 3. x ≥ y x is greater than or equal to y. 4. x ≤ y x is less than or equal to y. So, for example, x + 3 ≤ 2x may be read as “x + 3 is less than or equal to 2x.” Similarly, y > 0 is another way of saying “y is greater than 0.” Inequalities may also be written in compound form, such as 4 < y – 7 < 3y – 10. This is really just two separate inequalities: 4 < y – 7 and y – 7 < 3y – 10. Another way to think about this compound inequality is that y – 7, the expression stuck in the middle, is between 4 and 3y – 10. Solving Inequalities Solving inequalities is a lot like solving equations: Get the variable on one side of the inequality and all the numbers on the other, using the algebraic rules you’ve already learned. The one exception to this, and it’s a crucial exception, is that multiplying or dividing both sides of an inequality by a negative number requires that you flip the direction of the inequality. This exception is crucial, so we’ll repeat it: The Inequality Exception Multiplying or dividing both sides of an inequality by a negative number requires you to flip the direction of the inequality. Let’s try some examples. Solve for x in the inequality . First we knock that 3 away from the x by adding 3 to both sides: That 2 in the denominator is quite annoying, but by now you should know the fix for that—just multiply both sides by 2 and it will disappear from the left side of the inequality: Almost there. We can simplify the right side with our handy distributive law, multiplying the 2 by both terms in the parentheses to get a final answer of: We’re done. The x stands alone, and we know that it’s less than the expression 4y + 6. Now try this one: Solve for x in the inequality . Here are the steps, all at once: Notice that in this example the inequality had to be flipped, since both sides had to be divided by –2 to isolate the variable in the end. To help remember that multiplication or division by a negative number reverses the direction of the inequality, remember that if x > y, then –x < –y. Just as 5 is greater than 4, –5 is less than –4. The larger the number, the smaller it becomes when you make it negative. That’s why multiplying or dividing inequalities by negatives requires switching the direction of the inequality sign. Inequalities with Two Variables Another type of inequality problem involves two variables. For these, you’ll be given a range of values for each of the two variables. For example: –8 ≤ a ≤ 0 5 ≤ b ≤ 25 This is just another way of saying that a is between –8 and 0, inclusive, and that b is between 5 and 25, inclusive. Inclusive means that we include the values at each end, which is what’s meant by the greater than or equal to and less than or equal to signs. If the test makers didn’t want –8 and 0 to be possible values for a, for example, they would have to use simple greater-than and less-than signs (> and –x > –8. Absolute Value and Single Ranges Absolute values do the same thing to inequalities that they do to equations. You have to split the inequality into two parts, one reflecting the positive value of the inequality and one reflecting the negative value. You’ll see an example just below. If the absolute value is less than a given quantity, then the solution will be a single range with a lower and an upper bound. An example of a single range would be the numbers between –5 and 5, as seen in the following number line: A single range question will look something like this: Solve for x in the inequality |2x – 4| ≤ 6. First, split the inequality into two. In keeping with the rule for negative numbers, you’ll have to flip around the inequality sign when you write out the inequality for the negative scenario: 2x – 4 ≤ 6 2x – 4 ≥ –6 Solve the first: 2x – 4 ≤ 6 2x ≤ 10 x ≤ 5 Then solve the second: 2x – 4 ≥ –6 2x ≥ –2 x ≥ –1 So x is greater than or equal to –1 and less than or equal to 5. In other words, x lies between those two values. So you can write out the value of x in a single range, . Absolute Value and Disjointed Ranges You won’t always find that the value of the variable lies between two numbers. Instead, you may find that the solution is actually two separate ranges: one whose lower bound is negative infinity and whose upper bound is a real number, and one whose lower bound is a real number and whose upper bound is infinity. Yeah, words make it sound confusing. A number line will make it clearer. An example of a disjointed range would be all the numbers smaller than –5 and larger than 5, as shown below: On the GRE, disjointed ranges come up in problems in which the absolute value is greater than a given quantity, such as the following: Solve for x in the inequality |3x + 4| > 16. You know the drill. Split ’er up, then solve each inequality: 3x + 4 > 16 3x + 4 < –16 Again, notice that we have to switch the inequality sign in the second case because of the negative-number rule. Solving the first: 3x + 4 > 16 3x >12 x > 4 And the second: 3x + 4 < –16 3x < –20 Notice that x is greater than the positive number and smaller than the negative number. In other words, the possible values of x don’t lie between the two numbers; they lie outside the two numbers. So you need two separate ranges to show the possible values of x: –∞ < x < – and 4 < x < ∞. There are two distinct ranges for the possible values of x in this case, which is why the ranges are called disjointed. It doesn’t mean they can bend their fingers back all the way—that’s double-jointed. Made-up Symbols As if there aren’t enough real math symbols in the world, the GRE test makers occasionally feel the need to make up their own. You may see a made-up symbol problem on the GRE involving little graphics you’ve never seen before in a math context. Sure, they look weird, but they often involve variables and equations, which is why we thought we’d cover them here in the algebra section. Anyway, there’s a silver lining to this weirdness: Made-up symbol problems always give you exact instructions of what to do. Follow the instructions precisely, and you’ll do just fine. Let’s see how this works with an example: If x Ω y = 5x + 2y – 10, what is 3 Ω 4? Some test takers will see a problem like this and think, “Ω is one of those symbols that only math geniuses learned. I never learned this symbol. I’m not a math genius. I majored in Spanish. AAAAGGGHHHH!!!!” Relax: The fact is, no one learned what this symbol means, because the test makers made it up. Fortunately, the test makers have also made up a definition for it—that is, they tell us exactly what do to when we see a Ω. All it requires is some simple calculating. Look at the first part of the problem again: x Ω y = 5x + 2y – 10. All that’s really saying is that whenever we have two numbers separated by a Ω, we need to take 5 times the first number and 2 times the second number, add them together, and subtract 10. That’s it—these are the only instructions we need to follow. For the expression 3 Ω 4, then, x is 3, which we multiply by 5 to get 15. Similarly, y is 4, which we’re told to multiply by 2, which gives us 8. Adding these together gives us 23, and subtracting 10 brings us to 13 and a quick and easy point. Quick and easy, that is, if you don’t panic and just follow the directions to a T. Try one more example to get the hang of this symbol business: If j@k = , what is 5@–1? Forget the actual variables for a moment and focus on the instructions given: Whenever we have two numbers separated by an @, we need to divide the second number by 2, multiply the first number by 13 and subtract the two numbers. So, substituting 5 for j and –1 for k: The test makers may keep the choices as fractions, or they may decide to write them in decimal form, in which case –65.5 will be correct. Geometry-Algebra Hybrids: Coordinate Geometry Once upon a time, geometry and algebra hooked up after a drunken night. The result was coordinate geometry: a geometry-algebra hybrid. Coordinate geometry combines the graphical figures found in geometry with the variables used in algebra. So where does it belong, in algebra or geometry? Geometry wins on the name front, since it appears in the title and algebra doesn’t. But the GRE Official Guide includes this as an algebra topic, so we’ll defer to that and cover it here. The Coordinate Plane The coordinate plane is where all the magic happens. It’s the space in which coordinate geometry exists. Pretty snazzy. Every point on a coordinate plane can be mapped by using two perpendicular number lines. The horizontal x-axis defines the space from left to right. The vertical y-axis defines the space up and down. And the two meet at a point called the origin. Every point on the plane has two coordinates. Because it’s the center of the plane, the origin gets the coordinates (0,0). The coordinates of all other points indicate how far they are from the origin. These coordinates are written in the form (x, y). The x-coordinate is the point’s location along the x-axis (its distance either to the left or right of the origin). If a point is to the right of the origin, its x-coordinate is positive. If a point is to the left of the origin, its x-coordinate is negative. If a point is anywhere on the y-axis, its x-coordinate is 0. The y-coordinate of a point is its location along the y-axis (either up or down from the origin). If a point is above the origin, its y-coordinate is positive, and if a point is below the origin, its y-coordinate is negative. If a point is anywhere on the x-axis, its y-coordinate is 0. So the point labeled (2, 2) is 2 units to the right and 2 units above the origin. The point labeled (–7, –5) is 7 units to the left and 5 units below the origin. Quadrants Each graph has four quadrants, or sections. The upper-right quadrant is quadrant I, and the numbering continues counterclockwise. At the exact center of the graph, the origin is not in any of the four quadrants. Similarly, the x-axis and y-axis are not in any quadrant, either. You may be asked where a particular point exists. Although you could graph the point on a sketched graph, then use the diagram above, a faster method is to memorize the chart below: x-coordinate y-coordinate quadrant positive positive I negative positive II negative negative III positive negative IV This chart makes coordinate-location questions a breeze. Check it out: In which quadrant is coordinate pair (–8.4, –2) located? Sure, you could draw the coordinate plane and plot the pair. Or you could simply note that both the x-coordinate and the y-coordinate are negative, which means that (–8.4, –2) lives in quadrant III. Distance on the Coordinate Plane You may come across a GRE math question that asks you to find the distance between two points on the coordinate plane or to find the midpoint between two points. This news should make you happy. Why? Because these are fairly easy as long as you know the necessary formulas. There are two methods for finding distance and a formula for finding midpoints. We’ll tackle distance first. Finding Distance Using the Distance Formula If you know the coordinates of any two points—we’ll call them (x1, y1) and (x2, y2)—you can find their distance from each other with the aptly named distance formula: Let’s say you were suddenly overcome by the desire to calculate the distance between the points (4,–3) and (–3, 8). Just plug the coordinates into the formula: Finding Distance Using Right Triangles You can also solve distance questions using right triangles. Consider this one: What is the distance between the points (2, 4) and (5, 0)? Plot the two points on a graph and connect them with a straight line: The distance between these two points is the length of the dark line connecting them. To calculate this length, let the dark line be the longest side of a right triangle. Then draw two perpendicular lines to construct the short sides, as indicated by the slashed lines in the diagram. Since the horizontal side went from an x-coordinate of 2 to an x-coordinate of 5, its length is 5 – 2 = 3. Since the vertical side went from a y-coordinate of 0 to a y-coordinate of 4, its length is 4 – 0 = 4. These two sides fit the {3, 4, 5} right-triangle pattern, and so the long side must be 5. And that’s the distance between the two points. (No doubt you’ve learned about right triangles, but don’t worry, we’ll review them extensively in the upcoming geometry section.) So which method for calculating distance should you use? If you’re given the two points, there’s no reason why you shouldn’t be able to plug them into the distance formula and get the answer without bothering to sketch out a picture. The reason we also teach you the right-triangle method is because, hey, you may just like it better, but more important, a question may present you with a diagram to start, in which case the triangle method may be easier. Finding Midpoints To find the midpoint between points (x1, y1) and (x2, y2) in the coordinate plane, use this formula: In other words, the x- and y-coordinates of the midpoint are simply the averages of the x- and y-coordinates of the endpoints. (Again, we know you’ve heard of and worked with averages before, but we’ll review the concept in the data analysis section later in the chapter.) Applying the formula, the midpoint of the line in the coordinate plane connected by the points (6, 0) and (3, 7) is: Slope Coordinate geometry isn’t all the fun and games of calculating distances and naming quadrants—it’s also about finding slopes and solving equations. What follows isn’t particularly difficult, but terms such as slope and y-intercept rarely evoke positive feelings. Stay with us, and you’ll be okay. A line’s slope is a measurement of how steeply that line climbs or falls as it moves from left to right. The slopes of some lines are positive; the slopes of others are negative. Whether a line has a positive or negative slope is easy to tell just by looking at a graph of the line. If the line slopes uphill as you trace it from left to right, the slope is positive. If a line slopes downhill as you trace it from left to right, the slope is negative. Uphill = positive. Downhill = negative. You can get a sense of the magnitude of the slope of a line by looking at the line’s steepness. The steeper the line, the greater the slope; the flatter the line, the smaller the slope. Note that an extremely positive slope is larger than a moderately positive slope, while an extremely negative slope is smaller than a moderately negative slope. Check out the lines below and try to determine whether the slope of each line is negative or positive and which has the greatest slope: Lines a and b have positive slopes, and lines c and d have negative slopes. In terms of slope magnitude, line a > b > c > d. Slopes You Should Know by Sight There are certain easy-to-recognize slopes that it pays to recognize by sight: • A horizontal line has a slope of zero. • A vertical line has an undefined slope. • A line that makes a 45º angle with a horizontal line has a slope of either 1 or –1, depending on whether it’s going up or down from left to right. Of the four lines pictured below, which has a slope of 0, which has a slope of 1, which has a slope of –1, and which has an undefined slope? Line a has slope 0 because it’s horizontal. Line b has slope –1 because it slopes downward at 45º as you move from left to right. Line c has slope 1 because it slopes upward at 45º as you move from left to right. The slope of line d is undefined because it is vertical. Calculating Slope If you want the technical jargon, slope is a line’s vertical change divided by its horizontal change. Or, if you prefer the poetic version, slope is “the rise over run.” Okay, it’s not Hemingway (The Sun Also Rises), but it’ll do. For some strange reason, slope is symbolized by the letter m. We think s would be a better choice. Hello . . . Maybe s was taken. Anyway, you have two points on a line—once again (x1, y1) and (x2, y2)—the slope of that line can be calculated using the following formula: Rise is how far the line goes up between two points. Run is how far the line goes to the right between two points. The formula is just shorthand for the difference between the two y-coordinates divided by the difference between the two x-coordinates. Let’s work out an example: Calculate the slope of the line passing through (–4, 1) and (3, 4). The difference in y-coordinates, y2 – y1, is 4 – 1 = 3. The difference in x-coordinates, x2 – x1, is 3 – (–4) = 7. The slope is simply the ratio of these two differences, . Slopes of Parallel and Perpendicular Lines The slopes of parallel and perpendicular lines exhibit the following relationships: • The slopes of parallel lines are always the same. If one line has a slope of m, any line parallel to it will also have a slope of m. • The slopes of perpendicular lines are always the opposite reciprocals of each other. A line with slope m is perpendicular to a line with a slope of . In the figure below, lines q and r both have a slope of 2, so they are parallel. Line s is perpendicular to both lines q and r, so it has a slope of . Finding the Equation of a Line Now that you’ve mastered slope, you’re ready to determine the entire equation for a line that passes through two points. There are two ways to do this, and we’ll cover both. The Point-Slope Formula The equation of a line passing through two points in the coordinate plane can be expressed by the point-slope formula: (y – y1) = m(x – x1) You already know how to calculate the slope, m. And x and y are just variables. You don’t have to do anything special with them in the formula other than write them down, unchanged. As for x1 and y1, these are just the x- and y-coordinates of one of the points given. You can use whichever point you like; just make sure you use the same point for both the x1 and y1 values. Let’s use the formula to work through an example, using the same points we used in the previous question: What is the equation of the line passing through (–4, 1) and (3, 4)? We already calculated . Now we have to decide which of the two points to use in the equation. Again, either point will do. Maybe (3, 4) in this case would be a little easier, since it doesn’t have a negative sign. Letting x1 = 3 and y1 = 4 gives: Believe it or not, we’re almost done! We’ve actually written the equation of the line passing between the two points; all that’s left is to simplify it. First, eliminate the fraction by multiplying both sides by 7: 7(y – 4) = 7 (x – 3) 7y – 28 = 3(x – 3) Then distribute the 3 on the right side: 7y – 28 = 3x – 9 Finally, move the 3x to the left and the –28 to the right to isolate the variables: –3x + 7y = 19 This is the final equation of the line passing through (–4, 1) and (3, 4), written in simplest form. x-intercept and y-intercept The x- and y-intercepts are the points at which the graph intersects the x-axis and the y-axis, respectively. A useful way to think about this is that at the x- intercept, y = 0, and at the y-intercept, x = 0. Graphically, this is what the x- and y-intercepts look like for the equation we derived above: To calculate the precise values of these intercepts, recall the equation of this line, which we figured out earlier: –3x + 7y = 19. Since x = 0 at the y-intercept, you can calculate it simply by setting x = 0 in the equation, like this: 0 + 7y = 19 y = So is where the graph intersects the y-axis. Since the x-coordinate at this point is 0, the coordinate pair for the y-intercept is . Following the same mysterious naming conventions that cause slope to be named m, the point where the graph intersects the y-axis is called b. For the x-intercept, let y = 0 in the equation –3x + 7y = 19. This gives: –3x + 0 = 19 x = So is where the graph intersects the x-axis. Since the y-coordinate at this point is 0, the coordinate pair for the x-intercept is . The Slope-Intercept Equation The second method for calculating the equation of a line designated by two points in the coordinate plane is the slope-intercept equation. It looks like this: y = mx + b It’s likely that you had this one drilled into your head in high school, and you therefore recall the sound of it while having absolutely no idea what it means. Relearning it now is like being reacquainted with an old friend. Aren’t you glad you’re taking the GRE? You’re now familiar with each part of this equation: • x and y are variables, which you don’t need to change. • m is the slope, calculated by . • b is the y-intercept, the point where the graph intersects the y-axis. For the line we’ve been working with, the slope, m, is . The y-intercept, b, is . Plugging these values into the slope-intercept equation gives: This is the equation in slope-intercept format. You could simplify it by multiplying both sides by 7, then moving the variables to the left and the numbers to the right. As you may or may not expect, this gives exactly the same result as the point-slope format above: –3x + 7y = 19 This actually makes a lot of sense: The equation for the line should be the same, regardless of the method you use to calculate it. Which Equation Rules? You might be wondering when you should use each of the two equations, (y – y1) = m(x – x1) and y = mx + b. Decisions, decisions. Although slope-intercept is more commonly taught (most of us have at least heard of y = mx + b), the point-slope format is actually more useful when it comes to certain problems on the GRE. Both equations require the slope, so they’re similar in that respect. The slope-intercept formula requires a very specific point, the y-intercept, but the point-slope formula can be calculated with any point. Since it’s less restrictive, you may therefore find more uses for point-slope. But if they flat-out give you the slope and the y-intercept, then y = mx + b may be the way to go. The bottom line: Know both, and decide which to use depending on the question’s parameters. Word Problems Perhaps you’ve had nightmares in high school, certainly around SAT time, involving two trains, traveling in opposite directions, at different speeds, over different distances . . . That might be the point where you woke up screaming, or your dream morphed into Aunt Mabel offering you a piece of cheesecake. Well, relax. Word problems are nothing more than algebra problems, complicated by one small inconvenience: You have to come up with your own equations. Word problems present particular scenarios; the trick is to translate the information into math. The good news is that once you have your equations, they usually aren’t hard to solve. Pretty much any math concept is fair game for a word problem, but there are certain types that show up with some regularity. We’ll work through a few examples here, and you’ll see more later in the book. Using Simultaneous Equations Try this: In a sack of 50 marbles, there are 20 more red marbles than blue marbles. All of the marbles in the sack are either red or blue. How many blue marbles are in the sack? Well, these are very nice words and all, but talk is cheap—we want equations to crunch. And to get equations, we’re going to need to round ourselves up some variables. We don’t need a variable for total marbles, since that’s given: 50. However, we do need variables for the red and blue marbles, since those are unknown quantities. Unlike those nutty folks who come up with things like m for slope and b for y-intercept, we’ll be reasonable and just use r for red and b for blue. Since there’s no y-intercept for miles around in this one, we doubt anyone will mind if we borrow that b. Now for some translating: Is there a math expression that means the same thing as “there are 20 more red marbles than blue marbles”? You bet: r = b + 20 Since both r and b are unknown, we can’t solve the problem from this equation alone, so we’ll need to squeeze another equation out of this scenario. The only other facts are that there are 50 marbles total, and only red and blue ones in the sack. So it must also be true that: r + b = 50 Eureka! Why so excited? Because we have two different equations, each containing two variables, which means we have enough information to solve the problem. And if you remember from earlier in the chapter, we even have a choice of how to solve such simultaneous equations. We’ll go with substitution. Since the first equation tells us that r = b + 20, we can go ahead and substitute b + 20 for r in the second equation, giving us: b + 20 + b = 50 2b + 20 = 50 2b = 30 b = 15 Done. Now perhaps the CAT software would present you with the problem in the form we just solved. Or, perhaps, if you answered one or two more questions correctly before this one, that wily CAT would up the ante and make the question a tad harder by requiring an additional step. For example, instead of merely asking you to find the number of blue marbles in the sack, the question could ask you to find the ratio of blue to red marbles. No problem; it requires a few more steps, but it’s not so difficult. First, use either equation to find the number of red marbles. We’ll just plug 15 for b into r = b + 20, to get 35 reds. Then set up the ratio as 15:35 and simplify to 3:7. Ratios And speaking of ratios, here’s another kind of word problem based on this concept. In a certain class, the ratio of boys to girls is 4 to 5. If the class has 12 boys, what is the total number of boys and girls in the class? Again, the first step is to translate the English into algebra. Since ratios are part-to-part fractions, the ratio of boys to girls in the class is really just the fraction . A common careless mistake is to write the fraction upside down—that is, as . A great way to avoid making this mistake is to remember . In other words, whatever quantity appears after the word of in the problem goes in the numerator, and whatever quantity appears after the word to goes in the denominator. So, the ratio of boys to girls is 4 to 5, which, in math terms, can be written like this: We’re also told that the class has 12 boys. Substituting this into our equation gives: We’ll now use the Magic X to cross-multiply diagonally and up: 5 × 12 = girls × 4 60 = girls × 4 = girls 15 = girls Think 15 is the answer? Think again. The question didn’t ask for the number of girls, but for the total number of boys and girls in the class. Easy: Total = 12 + 15 = 27, and we’re done. Distance Since practically everyone who ever attended high school has had that nightmare about the trains traveling in different directions, the GRE test makers figure it’s a good problem to torment people with. Of course, they don’t always use trains. But these problems are really not that bad if you learn how to handle them; in fact, distance problems require only one fairly basic formula: rate × time = distance The concept is fairly intuitive: If you bike at a rate of 10 miles per hour, and you bike for 2 hours, you’re going to cover 10 × 2 = 20 miles. What gets confusing is the various ways they state these problems, but rest assured they always give you enough information to set up an equation and solve for the variable you seek. Here’s an example: Jim roller-skates 6 miles per hour. One morning, Jim roller-skates continuously for 60 miles. How many hours did Jim spend roller-skating? One thing that makes word problems annoying is that they sometimes include unnecessary facts. Here, for example, we’re told such exciting things as how Jim is traveling (by roller skates) and when he started (in the morning). Heck, it doesn’t even matter that his name is Jim—he could be Egbert the marble collector for all we care. You have to ignore the nonessentials and focus on the facts you need to solve the problem. Begin with the trusty distance formula: rate × time = distance Then fill in the values you know: 6 miles per hour × time = 60 miles Divide both sides by 6 miles per hour to calculate the time spent as 10 hours. This next distance problem is a bit more difficult because it requires one extra step. Instead of being given two variables and simply setting up an equation to solve for the third, you need to first calculate one of the variables before getting to that point. Try it out: A traveler begins driving from California and heads east across the United States. If she drives at a rate of 528,000 feet per hour, and drives 4.8 hours without stopping, how many miles has she traveled? (1 mile = 5,280 feet) The reason you need to go the extra mile (so to speak) is because the rate is given in feet per hour but the distance traveled is given in miles. The units must jibe for the problem to work, so a conversion is necessary. Luckily, the numbers are easy to work with: The driver drives 528,000 feet per hour, and 5,280 feet equal one mile. So to change her rate into miles per hour, simply divide: = 100 miles per hour That’s some fast traveler. Now plug the values from the problem, including this new one, into the formula: 100 miles per hour × 4.8 hours = distance The distance traveled is 480 miles. Work Work sucks. You’re there from at least 9 to 5, get two weeks off per year (at best), and you’ve got a boss constantly checking up on you. Work word problems on the GRE are a breeze in comparison. Work word problems are very similar to distance word problems, except the final outcome is units of something produced instead of distance covered. The other two variables are analogous to those in distance problems: the rate at which the units are produced, and the time someone spends producing them. If you knit 2 sweaters per hour, and knit for 8 hours, then you’ll knit 2 × 8 = 16 sweaters. This generalizes to the same basic formula we saw earlier: rate × time = units produced Since you’ve seen this basic mechanism in action in the previous section, we’ll jump right to a difficult work word problem: Four workers can dig a 40-foot well in 4 days. How long would it take for 8 workers working at the same rate to dig a 60-foot well? Sure, it’s complicated, but the three parameters are the same: rate, time worked, and amount of work completed. We know the latter: 60 feet dug. The middle amount, time, is what we’re looking for, so we’ll just leave that as our unknown variable for now. The question then hinges on the rate, which we need to calculate ourselves. Well, since 4 workers dig 40 feet in 4 days, we can divide by 4 to determine that those 4 workers can dig 10 feet in one day. The 8 workers that we’re interested in dig at the same rate, and since they represent twice as many people, they’ll be able to dig twice as many feet in a day: 20. Now we have the variables to plug into our formula: 20 feet per day × time = 60 units produced (in this case, feet dug) Divide both sides by 20 to get a time of 3 days of work for the 8 workers to dig the 60-foot well. Many other word problems follow the same basic form as the examples above. For example, in a price word problem, the price of an item times the number of items bought will equal the total cost. The particulars are slightly different, but the concept is basically the same. Use the information given in the problem to get two of the values, and you’ll always be able to calculate the third. And you’ll never have a word problem nightmare again. That wraps it up for algebra. Now we move on to triangles, circles, and all of the other fun shapes of our next math subject, geometry. > Help | Feedback | Make a request | Report an error
Posted on: Sat, 22 Mar 2014 09:36:15 +0000
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