Chapter 30 Algebra quantitative analysis

CERTAIN PROPERTIES Finally, variables may be defined as having certain properties. The most common include a variable being positive or negative or an integer. The strategy for this type of problem is prove (D). x is positive. Quantity A ___________x(x + 1) Quantity B ___________x(x ^2 + 1)

Begin by distributing both quantities: x is positive. Quantity A ____x(x + 1) = x 2 + x Quantity B___ x(x ^2 + 1) = x^ 3 + x Both sides have an x, which you can cancel out. You know x is positive, so x can't be negative or 0. If x = 2, then Quantity A = 4 and Quantity B = 8. To be thorough, however, make sure that you try numbers that have a chance of behaving differently. You can't try negatives, but you can try 1 and fractions between 0 and 1. If x = 1, then Quantity A = 1 and Quantity B = 1. Also, if x were a positive fraction (e.g., 1/2), then Quantity A would be greater than Quantity B. The correct answer is (D).

RELATIVE ORDER Example 1 0 < p < q < r Quantity A__________ p + q Quantity B______________ q + r Example 2 0 < p < q < r Quantity A _______ pq Quantity B _______ qr Example 3 0 < p < q < r Quantity A _______ q − p Quantity B________ r − q Example 4 0 < p < q < r Quantity A __________z/y Quantity B____________r/q

Example 1 ans= The correct answer is a (B). Example 2 ans= The correct answer is a (B). Example 3 ans= The correct answer is a (D). Example 4 Because all the variables are positive, you can cross-multiply: Quantity A _____q ^2 Quantity B______ pr It's impossible to know for sure which quantity will be bigger. For extra practice, use numbers satisfying 0 < p < q < r to prove the answer is (D). (Hint: Space the numbers differently, as in the previous example.) Strategy Tips: When absolute values contain variables, maximize the absolute value by making the expression inside as far away from 0 as possible. 1) If the absolute value also contains a positive number, make the variable positive to maximize the absolute value .2) If the absolute value also contains a negative number, make the variable negative to maximize the absolute value Sometimes inequalities are used to order variables from least to greatest.gave the sign of the variables, and gave their order from least to greatest. To compare the two quantities, use the Invisible Inequality to eliminate common terms, and try to discern a pattern if one is present

Strategy Tips: Variables are used in many ways on this test. How they're presented can often give you a clue as to the appropriate strategy to employ. To recap:

If a variable: _____________________________then: has a unique value (e.g., x + 3 = −5) then solve for the value of the variable has a defined range (e.g., −4 ≤ w ≤ 3) then test the boundaries has a relationship with another variable (e.g., 2p = r) then simplify the equation and make a direct comparison of the variables has no constraints then try to prove (D) has specific properties (e.g., x is negative) then try to prove (D)

x − 3 = 12 y + 2x = 40 Quantity A________ y Quantity B_________ 9

Problem Recap: When variables have a unique value, you must solve for the value of the variable. Pay attention to any constraints that have been placed on variables. In this question, the first equation gives you enough information to find the value of x. And because you have enough information to find x, you also have enough information to find y through the second equation. In fact, x = 15, which means that y will equal 10, and thus the answer to this question is (A). Although y is a variable, it actually has a definite value

RELATIONSHIP ONLY Another way variables can be defined on this test is in terms of another variable. Take the following example: X + 5/5 =Y + 6/6 Quantity A ________ 6x Quantity B________ 5y

Problem Recap: If a variable is defined in terms of another variable, simplify and find a direct comparison. Begin by cross-multiplying: 6(x+5)= 5(Y+6) 6x+30=5y+30 Now you have a 30 on each side that should be eliminated: 6x = 5y You still don't know the value of either variable, but you do have enough information to answer the question. The answer is (C

2 ≤ z ≤ 4 Quantity A _______337,0000/1000,000 Quantity B__________5/2(2)= 5/4

Strategy Tip: If a variable has a defined range, you need to test the boundaries of that range When z = 2, Quantity B is bigger. Now try the upper bound. Plug in 4 for z in both quantities: When z = 4, Quantity A is bigger. The correct answer is (D). The way in which variables are constrained (or not) can tell you a lot about efficient ways to approach that particular problem

NO CONSTRAINTS Sometimes, you will not be given any information about a variable. If there are no constraints on the variable, then your goal is to prove (D). For example: Quantity A ____________x/2 Quantity B____________ 2x

Strategy Tip: If a variable has no constraints, try to prove (D). No information about x has been given. If x is positive, Quantity B will be bigger. For instance, if x = 1, Quantity A=1/2 Quantity B=2 However, there is no reason x must be positive. Remember, one way to try to prove (D) is to check negative possibilities. If x is negative, then Quantity A will be bigger. For instance, if x = −1, then Quantity A=-1/2 and Quantity B= -2

As QC questions involving quadratic expressions get more difficult, they can make either FOILing or simplifying more difficult. Try this example problem r > s Quantity A _____________(r + s)(r − s) Quantity B_______________ (s + r)(s − r)

Strategy Tip: The challenging part of this question was comparing the quantities aer you had FOILed them. Notice you had to incorporate knowledge of positives and negatives to come to the correct conclusion. Harder questions will be difficult for either of two reasons:1. Expressions are hard to FOIL, or 2. the comparisons are challenging This problem now requires you to FOIL two expressions, r > s Quantity A (r + s)(r − s) = r ^2 − s^ 2 Quantity B (s + r)(s − r) = s ^2 − r ^2 Now you need to be able to compare these expressions. You know r is greater than s, so it might be tempting to conclude that Quantity A is greater than Quantity B. Plug in r = 3 and s = 2:Quantity A is greater than Quantity B. But there's a problem. You know r is greater than s, but you don't know the sign of either variable. Remember to check negative possibilities! Now plug in r = −2 and s = −3: Here, you get the opposite conclusion, that Quantity B is greater than Quantity A. Because you can't arrive at a consistent conclusion, the answer is (D).

QUADRATICS IN QUANTITIES If the quadratic expressions appears in the quantities, then your goal is to FOIL and eliminate common terms to make a direct comparison. pq ≠ 0 Quantity A _______(2p + q)(p + 2q) Quantity B _______p ^2 + 5pq + q ^2

Strategy Tip: When a quadratic expression appears in one or both quantities, FOIL the quadratics, eliminate common terms, and compare the quantities. . To answer this question correctly, you had to do two things: 1) FOIL Quantity A (the faster the better), and 2) eliminate common terms from both quantities and compare the remaining terms. The correct answer is (A).

−2 ≤ x ≤ 3 −3 ≤ y ≤ 2 Quantity A The maximum value of | x − 4| Quantity B The maximum value of | y + 4

Strategy Tip: When absolute values contain a variable, maximize the absolute value by making the expression inside as far away from 0 as possible. Add positives to positives or add negatives to negatives Once again, inequalities are used to bound a variable. As before, you should test the boundaries of the range. But now, there's the added twist of absolute values. On QC, it is important to understand how to maximize and minimize values. The smallest possible value of any absolute value will be 0. This question asks you to maximize the absolute values in Quantities A and B. In Quantity B, the maximum value of | y + 4| will be when y = 2, because that is the largest number you can add to positive 4. The absolute value of | y + 4| will equal 6 To maximize the absolute value of |x − 4| in Quantity A, however, you have to do the opposite. There is a negative 4 already in the absolute value. If you try to increase the value by adding a positive number to −4, you will only make the absolute value smaller. For instance, if x is 3, then the absolute value is: |3 − 4| = |−1| = 1 You can actually maximize the absolute value by making x = −2. Then the absolute value becomes: |−2 − 4| = |−6| = 6 The maximum value in each quantity is the same (6), therefore, the answer is (C).

QUADRATICS IN COMMON INFORMATION Questions that contain quadratic equations in the common information will present different challenges. For example: x ^2 − 6x + 8 = 0 Quantity A _________ x^ 2 Quantity B___________ 2 x

The first thing to note here is that there will be two possible values for x. But you should not jump to conclusions and assume the answer will be (D). To make sure you get the right answer, you need to solve for both values of x and plug them both into the quantities. First, solve for x by factoring the equation so that it reads (x − 2)(x − 4) = 0. That means that x = 2 or x = 4. Start by plugging in 2 for x in both quantities When x = 2, the quantities are equal. Now try x = 4: Even though there are two possible values for x, both of these values lead to the same conclusion: the quantities are equal. The correct answer is (C). Strategy Tip: When the common information contains a quadratic equation, solve for both possible values and put them into the quantities.Section Recap There are two types of questions involving quadratics.Each type will require a different approach .1. If a quadratic appears in one or both quantities:a. FOIL the quadratic,b. eliminate common terms, and c. compare the quantities. 2. If a quadratic appears in the common information:a. factor the equation and find both solutions, and b. plug both solutions into the quantities.

v& = 2v − 1 Quantity A______ (v&)& Quantity B_______ 4v

evaluate the formula using the variable itself. Start by evaluating the formula inside the parentheses: v& = 2v − 1 Rewrite Quantity A as (2v − 1)&. Evaluate the formula one more time.Now, no matter what v is, Quantity B will be bigger Once the formula was evaluated, a clear comparison could be made between the quantities. Strategy Tip:There are two types of questions involving strange symbol formulas. Each type will require a different approach 1) If the question contains numbers, plug in the numbers and evaluate the formula. 1. 2) If the question does not contain numbers, plug the given variable(s) directly into the formula −2 ≤ x ≤ 3