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Cracking the SAT Subject Test in Math 2, 2nd Edition math |a − 75|< 5 Understanding this way of thinking about ranges can be helpful in many questions Another approach to these questions is to Plug In[.]
math |a − 75|< 5 Understanding this way of thinking about ranges can be helpful in many questions Another approach to these questions is to Plug In If you Plug In on these questions, be sure to not only try different values which work given the conditions (eliminating answer choices which are not true), but also try values which NOT follow the given conditions (eliminating answer choices which are TRUE given the false values for the variable) DRILL 7: MORE WORKING WITH RANGES Try the following range questions The answers can be found in Part IV If −2 ≤ a ≤ 7 and 3 ≤ b ≤ 9, then what is the range of possible values of b − a ? If 2 ≤ x ≤ 11 and 6 ≥ y ≥ − 4, then what is the range of possible values of x + y ? If −3 ≤ n ≤ 8, then what is the range of possible values of n2 ? If 0 < x < 5 and −9 < y < −3, then what is the range of possible values of x − y ? If −3 ≤ r ≤ 10 and −10 ≤ s ≤ 3, then what is the range of possible values of r + s ? If −6 < c < 0 and 13 < d < 21, then what is the range of possible values of cd ? If |3−x|≤ 4, then what is the range of possible values of x ? If |2a+7|≥ 13, then what is the range of possible values of a ? DIRECT AND INVERSE VARIATION Direct and indirect variation are specific relationships between quantities Quantities that vary directly are said to be in proportion or proportional Quantities that vary indirectly are said to be inversely proportional Direct Variation If x and y are in direct variation, that can be said in several ways: x and y are in proportion; x and y change proportionally; or x varies directly as y All of these descriptions come down to the same thing: x and y increase and decrease together Specifically, they mean that the quantity will always have the same numerical value That’s all there is to it Take a look at a question based on this idea A Great Way to Remember To remember direct variation, think “direct means divide.” So in order to solve, you set up a proportion with a fraction on each side of the equation Just solve for the one number you don’t know There are two formulas associated with direct variation that may be familiar to you They are: or y = kx, where k is a constant If n varies directly as m, and n is 3 when m is 24, then what is the value of n when m is 11 ? (A) 1.375 (B) 1.775 (C) 1.95 (D) 2.0 (E) 2.125 Here’s How to Crack It To solve the problem, use the definition of direct variation: must always have the same numerical value Set up a proportion Solve by cross-multiplying and isolating n 24n = 33 n = 33 ÷ 24 n = 1.375 And that’s all there is to it The correct answer is (A) Inverse Variation If x and y are in inverse variation, this can be said in several ways as well: x and y are in inverse proportion; x and y are inversely proportional; or x varies indirectly as y All of these descriptions come down to the same thing: x increases when y decreases, and decreases when y increases Specifically, they mean that the quantity xy will always have the same numerical value Opposites Attract A great way to remember indirect or inverse variation is that direct and inverse are opposites What’s the opposite of division? Multiplication! So set up an inverse variation as two multiplication problems on either side of an equation There are two formulas associated with indirect variation that may be familiar to you They are: x1y1 = x2y2 or y = , where k is a constant Take a look at this question based on inverse variation: If a varies inversely as b, and a = when b = 5, then what is the value of a when b = 7? (A) 2.14 (B) 2.76 (C) 3.28 (D) 4.2 (E) 11.67 Here’s How to Crack It To answer the question, use the definition of inverse variation That is, the quantity ab must always have the same value Therefore, you can set up this simple equation So the correct answer is (A) DRILL 8: DIRECT AND INVERSE VARIATION Try these practice exercises using the definitions of direct and inverse variation The answers can be found in Part IV 2 If a varies inversely as b, and a = 3 when b = 5, then what is the value of a when b = x? (A) (B) (C) (D) 3x (E) 3x2 If n varies directly as m, and n = 5 when m = 4, then what is the value of n when m = 5 ? (A) 4.0 (B) 4.75 (C) 5.5 (D) 6.25 (E) 7.75 If p varies directly as q, and p = 3 when q = 10, then what is the value of p when q = 1 ? (A) 0.3 (B) 0.43 (C) 0.5 (D) 4.3 (E) 4.33 11 If y varies directly as x2, and y = 24 when x = 3.7, what is the value of y when x = 8.3? (A) 170.67 (B) 120.77 ... DRILL 8: DIRECT AND INVERSE VARIATION Try these practice exercises using the definitions of direct and inverse variation The answers can be found in Part IV 2 If a varies inversely as b, and a = 3 when b = 5, then what is the value of a when b = x?... And that’s all there is to it The correct answer is (A) Inverse Variation If x and y are in inverse variation, this can be said in several ways as well: x and y are in inverse proportion; x and y are inversely proportional; or x... use the definition of inverse variation That is, the quantity ab must always have the same value Therefore, you can set up this simple equation So the correct answer is (A) DRILL 8: DIRECT AND INVERSE VARIATION