ACT study guide Math: Inequalities and Absolute Value Equations

For which values of x will 3(x + 4) ≥ 9(4 + x)?

((Remember that the direction of the sign in an inequality switches when dividing by a negative number.) To answer this question, solve the inequality for x, as follows: 3(x+4)≥9(4+x) 3x+12≥36+9x −6x≥24) x≤−4

If |5 − 2x| > 5, which of the following is a possible value of x?

(Given that |5 − 2x| > 5, then either 5 − 2x > 5 or 5 − 2x < −5. In the case that 5 − 2x > 5, −2x > 0 making x < 0 (when you divide by a negative number remember to switch the direction of the inequality). In the case that 5 − 2x < −5, then −2x < −10 making x > 5. Thus the range for x is x < 0 or x > 5. Of the answer choices, only 6 fits into the range for x.) 6

If x^2−3≤13, what is the greatest real value that x can have?

(Here, you must first solve the inequality: x2−3≤13 x2≤16 Because a negative number squared will yield a positive result, you must consider both the positive and negative values of x. x≤4 and x≥−4. The greatest real value of x is) 4

If f (m, n) = m2 − n − 3 and both m and n are positive numbers, then which of the following statements must be true?

(If n is larger than m2, the expression will be negative. Otherwise, as long as m2 is larger than n + 3, the expression will be positive.) Cannot be determined from the given information

If x < y, then |x - y| is equivalent to which of the following?

(One way to solve this problem is to pick numbers for x and y. You are given that x < y, so make x = 2 and y = 3. Now, substitute those values into the absolute value given: |x - y| = |2 - 3| = | - 1| = 1. Now, substitute the values into each answer choice. The answer choice that yields a value of 1 will be correct: Answer choice A: 2 + 3 = 5; eliminate answer choice A. Answer choice B: −(2 + 3) = −5; eliminate answer choice B. Answer choice C: 2⎯⎯√ − 3 ≠ 1; eliminate answer choice C. Answer choice D: 2 − 3 = − 1; eliminate answer choice D Answer choice E: −(2 − 3) = −(−1) = 1. Answer choice E is correct. For any values of x and y when x < y, |x − y| = )− (x − y).

The temperature, t, in degrees Fahrenheit, in a certain city on a certain spring day satisfies the inequality |t − 34| ≤ 40. Which of the following temperatures, in degrees Fahrenheit, is NOT in this range?

(One way to solve this problem is to substitute each of the answer choices for t in the inequality to determine which one does NOT work: Answer choice A: |74 − 34| = |40| = 40; 40 ≤ 40, so eliminate answer choice A. Answer choice B: |16 − 34| = | − 18| = 18; 18 ≤ 40, so eliminate answer choice B. Answer choice C: |0 − 34| = | − 34| = 34; 34 ≤ 40, so eliminate answer choice C. Answer choice D: | − 6 − 34| = | − 40| = 40; 40 ≤ 40, so eliminate answer choice D. By the process of elimination, answer choice E must be correct: | − 8 − 34| = | − 42| = 42, which is NOT ≤ 40) -8

What value of a is a solution to the inequality a − x ≥ c but is NOT a solution to the inequality a − x > c?

(Solving for a in both inequalities gives us a≥c+x and a > c + x. The only difference between these is that a≥c+x allows for a to equal c + x, while a > c + x does not.) a = c + x

For what value(s) of y is x − y > x + y true for any possible value of x?

(Subtracting x from both sides of the inequality yields −y > y. This can be true only when )y < 0

For what value of b would the following system of equations have an infinite number of solutions? 3x + 5y = 27 12x + 20y = 3b

(Systems of equations have an infinite number of solutions when the equations are equivalent (i.e. they graph the same line). In order for the two equations to be equivalent, the constants and coefficients must be proportional. If the entire equation 3x + 5y = 27 is multiplied by 4, the result is 4(3x + 5y) = 4(27), or 12x + 20y = 108. Thus, in order for the two equations to be equivalent, 3b = 108, or b = )36

If 2m = 41, which of the following inequalities must be true?

(The closest values for whole number powers of 2 are 25 = 32 and 26 = 64. Since 41 is between 32 and 64, m must be between 5 and 6.) 5 < m < 6

If a + b = 25 and a > 4, then which of the following must be true?

(The correct answer will be the statement that is always true. Because a is greater than 4, and 25 − 4 = 21, b must always be less than 21) b < 21

Which of the following is a possible value of x if 4x > 3 and −x > 3?

(The first inequality is equivalent to x>34, while the second is equivalent to x < −3. There are no real numbers that are larger than 34 and at the same time smaller than −3) There are no values of x that satisfy both inequalities

If a set is defined as all numbers satisfying a given inequality, then which of the following sets would NOT share any values with the set of values for x satisfying −1 ≤ x < 5?

(The inequality given is representing the set of numbers between −1 and 5, including the −1 but not including the 5. The set in answer choice A does not include the −1 and contains only negative numbers.) −5 ≤ x < −1

What is the solution set of |3a − 2| ≤ 7?

(To find the solution set for |3a − 2| ≤ 7, break it up into two separate inequalities: 3a − 2 ≤ 7 and 3a − 2 ≥ −7. Starting with 3a − 2 ≤ 7, solving for a yields a≤3. With 3a − 2 ≥ −7, solving for a yields a≥−5/3. Thus a is between −5/3 and 3 inclusive)a:−53≤a≤3

For which of the following functions is f(−5) > f(5)?

(To solve this problem, first eliminate answer choices that yield equal values for f (−5) and f (5). These include answer choices in which the functions have even powers of x such as answer choice A, where f (x) = 6x2, answer choice B, where f (x) = 6, and answer choice E, where f (x) = x6 + 6. Now, substitute −5 and 5 into the remaining answer choices: Answer choice C: f (x) = 6x . When x is −5, f (x) = −6/5, and when x is 5, f (x) = 6/5. Therefore, f (5) is greater than f (−5) and answer choice C is incorrect. Answer choice D: f (x) = 6 − x3. When x = −5, f (x) = 6 − (−125) or 131, and when x is 5, f(x) = 6 − 125, or −119. Therefore, f (−5) is greater than f (5).) f(x) = 6 − x3

|5 − 3| − |2 − 6| = ?

(To solve this problem, first perform the subtractions and then evaluate the absolute values: |5−3|−|2−6| |2|−|−4| 2−4=)−2

What is the solution set of |3a − 3| ≥ 12?

(To solve this problem, remember that |3a − 3| ≥ 12 is equivalent to 3a − 3 ≥ 12 or 3a − 3 ≤ − 12. Adding 3 to both sides and dividing by 3 yields )a ≥ 5 or a ≤ − 3

For all values, x, y, and z, if x ≤ y and y ≤ z, which of the following CANNOT be true? I. x = z II. x > z III. x < z

(You are given that x ≤ y and y ≤ z. If x = y and y = z, then x = z could be true. Eliminate answer choices A, D, and E, which show that Roman numeral I (x = z) CANNOT be true. If x < y and y < z, then x < z could be true. But x > z could not be true in either case.) Only II CANNOT be true,

If 2p+q/p>5 and p > 2, Which of the following inequalities MUST be true?

Rewriting the first inequality shows that 2p + q > 5p. Collecting terms, q > 3p. Therefore, when p > 2, q > 6.

For which of the following values of x is the value of g(x) = −x2 + 10 negative?

Substituting each of the values, g(−4) = −6

The equation |x−y|<12 can be interpreted as "the distance between x and y is less than 1/2." On the provided number line, only points C and D are less than one-half unit apart

Using the following number line, which pairs of points satisfy the equation |x−y|<1/2?