ACT study guide Math: Linear Inequalities with One Variable

If p − 5 > −2, then which of the following inequalities must be true?

(All of the answer choices include the term p + 5. We can change p − 5 to p + 5 by adding 10, because p − 5 + 10 = p + 5. The cardinal rule of algebra is to do to one side what you have done to the other side, so once 10 has been added to the left side of the equation (p − 5 + 10 = p + 5), add 10 to the right side, too: −2 + 10 = 8. The resulting inequality is )p + 5 > 8

For what values of m is m2− 8m + 15 < 0?

(By factoring the given inequality, we see that it can be rewritten as (m − 5)(m − 3) < 0 and that the function has roots of 3 and 5. This means the function will be negative either before 3 (m < 3), after 5 (m > 5), or in between (3 < m < 5). By testing values, we can determine which is true. For values before 3, testing m = 0 yields 02 − 8(0) + 15 = 15 > 0. For values after 5, m = 6 yields 62 −8(6) + 15 = 3 > 0. Finally, for values between 3 and 5, m = 4 yields 42 − 8(4) + 15 = −1 < 0, showing us that the function is negative for all of these values.) 3 < m < 5

Which of the following inequalities has the same solution set as x − 8 ≥ 5x + 1?

(Each inequality in the answer choices has the reverse direction of the one given. This must mean both sides were multiplied by −1. When both sides are multiplied by −1, the signs on every term will change.) −x + 8 ≤ −5x − 1

Which of the following inequalities represents the set of all values of x that satisfy the inequality −3x + 4 > 6?

(Solving inequalities works the same way as solving equations. However, we must remember that multiplying or dividing both sides by a negative value will switch the order of the inequality. Therefore, when we simplify the inequality to −3x > 2, the solution becomes) x<−2/3

Which of the following represents the solution set to the inequality −1/2x+5≥9?

(Subtracting 5 from both sides of the inequality yields the inequality −1/2x≥4. To solve, multiply both sides by −2:)x≤−8

Considering all values of a and b for which a + b is at most 9, a is at least 2, and b is at least −2, what is the minimum value of b − a?

(The first step in solving this problem is to rewrite the information in mathematical terms, as follows: "a + b is at most 9" means that a + b < 9 "a is at least 2" means that a > 2 "b is at least − 2" means that b > − 2 Given the information above, the value of b − a will be least when b is at its minimum value of −2. In that case, since a + b ≤ 9, then a + (−2) ≤ 9, and a ≤ 11. Therefore, at its minimum, b − a is equivalent to −2 − 11, or )−13

If |6 - 2x| > 9, which of the following is a possible value of x?

(This problem tests your knowledge of absolute values and inequalities. The inequality |6 − 2x| > 9 can be written as two separate inequalities: 6 − 2x > 9 or 6 − 2x < −9. Because the original inequality was a greater than, the word in between the two new inequalities must be "or." This means that 6 − 2x > 9 = − 2x > 3 = x < − 3/2, or that 6 − 2x < − 9 = −2x < −15 = x > 15/2. (Remember to reverse your inequality when multiplying or dividing by negative numbers.) The only answer choice that fits one of the inequalities is −2, which is less than −3/2. There are no answer choices that are greater than 15/2) -2

Which of the following is the solution set of x + 2 > −4?

(To find the solution set of x + 2 > −4, first solve for x by subtracting 2 from both sides. The result is x > −6. Thus the solution set is x:) x > −6

What is the smallest integer for which the inequality x/4−1/4≥1 is true?

(To isolate the variable x, you must first add 1/4 to both sides of the equation, and then multiply both sides by 4. For the inequality to be true, x/4 must be more than 1/4 larger than 1 or simply equal to 1+1/4 Since 4/4 + 1, 4/4 + 1/4=5/4 will satisfy the greater than or equal to condition, so )x = 5

Which of the following inequalities defines the solution set for the inequality 23 − 6x ≥ 5?

(To solve this problem, first subtract 23 from both sides of 23 − 6x ≥ 5 to get −6x ≥ − 18. Then divide both sides of the inequality by −6 (flip the direction of the inequality when dividing by a negative number) to get) x ≤ 3

What is the smallest possible integer for which 15% of that integer is greater than 2.3?

(To solve this problem, make x the smallest possible integer for which 15% of x is greater than 2.3. Then, set up the following inequality: 0.15x > 2.3. Divide both sides by 0.15 to get x > 15.333, repeating. The smallest integer greater than the repeating decimal 15.333 is )16

If 3 times a number x is added to 12, the result is negative. Which of the following gives the possible value(s) for x?

(To solve this problem, write a mathematical expression for the phrase in the problem. Because when 3 times a number x is added to 12 the result is negative, 3x + 12 < 0. To solve for x, subtract 12 from both sides and divide by 3 to get ) all x < −4

How many different integer values of a/4 satisfy the inequality 1/11<2/a<1/8?

(You are asked for integer values of a, which means that a must be a whole number. The first step will be to change 111 to 2/22 and 1/8 to 2/16; now it is easy to see that if 2a is between 2/22 and 2/16, a can be equal to 21, 20, 19, 18, or 17. Only 20 is divisible by 4: 20/4 = 5. This is 1 value) 1

What is the complete solution to the following equation? 28+10x<4−2x

. 28+10x<4−2x 12x<−24 x<−2

If g(a, b) = 4ab − b + a, then for what values of b is g(1, b) > 0?

For a = 1, substitute and simplify: g(1, b) = 4b − b + 1 = 3b + 1 > 0. For the inequality where g(1, b) > 0, solve for b: 3b + 1 > 0, so b>−1/3

How many positive integers satisfy the inequality −5 ≥ −7x?

The given inequality is equivalent to the inequality 5/7≤x, and every positive integer is larger than 5/7

Each of the following integers satisfies the inequality −12≤5x≤332 EXCEPT

To solve for x, divide each term in the inequality by 5, and rewrite it as −1/10≤x≤33/10. Each value given is a solution to this inequality except 4, which is larger than 33/10