Cracking the SAT subject test in math 2, 2nd edition

Cracking the SAT Subject Test in Math 2, 2nd Edition INEQUALITIES Inequalities can be treated just like equations You can add, subtract, multiply, and divide on both sides of the inequality sign And y[.]

Inequalities can be treated just like equations You can add, subtract, multiply, and divide on both sides of the inequality sign And you still solve by isolating the variable There is one major difference between solving an equation and solving an inequality

Reading Inequality Signs

Here’s how you should read the four basic

inequality signs:

a < b a is less than b

a > b a is greater than b

a ≤ b a is less than or equal to b

a ≥ b a is greater than or equal to b

Whenever you multiply or divide both sides of an inequality by a negative, flip the inequality sign

Multiplying across an inequality by a negative flips the signs of all of the terms in the inequality The inequality sign itself must also flip

This rule becomes a little more complicated when you are multiplying or dividing by a variable For example, when you attempt to solve > 3, if

you multiply both sides by y you don’t know whether the result is x > 3y

or x < 3y because you do not know whether y is positive or negative Be

careful in these cases, as ETS likes to test this rule in questions which ask what “must be true.”

Similarly, when you solve an inequality with an even exponent or absolute value, remember to include both positive and negative roots In

an inequality, you need to flip the sign when expressing the negative root:

a2 ≥ 25 becomes or a ≥ 5 or a ≤ −5

y4 < 18 becomes −3< y < 3

|x + 2|> 5 becomes x > 3 or x < −7

Often Plugging In or Plugging In the Answers is the best way to deal with questions involving variables and inequalities

3 If x3 + 3x2 + 3x + 3 ≥ 2, then which of the following is

true?

(A) x ≠ 0 (B) x ≥ −1 (C) x ≤ −1 (D) x ≥ 2 (E) x ≤ 2

Here’s How to Crack It

You can Plug In to the inequality and then eliminate answer choices Pick

a number that is true for some answer choices but false for others Next, Plug In that number into the original inequality If the number makes the inequality true, eliminate answer choices that exclude that number If the number makes the inequality false, eliminate answer choices that include that number

Here, start with x = 3 3 is included in (A), (B), and (D), and excluded in (C) and (E) When you Plug In x = 3 in the original inequality, you get 33

+ 3(3)2 + 3(3) + 3 ≥ 2, or 66 ≥ 2 This is true, so x = 3 must be included in your answer Eliminate (C) and (E) Next, make x = 0 0 is included in (B) and excluded from (A) and (D) When you Plug In x = 0 into the original

inequality, you get 03 + 3(0)2 + 3(0) + 3 ≥ 2, or 3 ≥ 2 This is true, so x =

DRILL 5: INEQUALITIES

Practice solving inequalities in the following exercises The answers can

be found in Part IV

4 If 8(3x + 1) + 4 < 15, then

6 If 4n − 25 ≤ 19 − 7n, then

7 If −5(p + 2) < 10 p − 13, then

9 If −3x − 16 ≤ 2x + 19, then