ACT · Math · Preparing for Higher Math: Algebra

Inequalities

Inequalities is part of Preparing for Higher Math: Algebra in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include solve a linear inequality, compound inequality, inequality from a context and count integer solutions.

Common traps: not reversing the inequality for a negative; reversing the inequality when no negative is used; dropping the equal sign.

Sample questions

Question 1
What is the solution set of the inequality 5 − 4x > 21?
- A.
  x < −4
- B.
  x < −6.5
- C.
  x > −4
- D.
  x < 4
Answer: A. x < −4
Subtract 5: −4x > 16. Dividing by −4 reverses the sign, so x < −4. C does not reverse the inequality; D drops the negative on 16 ÷ (−4); B adds 5 instead of subtracting it (−4x > 26).
Question 2
Which inequality says that x is at least −3 and less than 7?
- A.
  −3 < x ≤ 7
- B.
  −3 < x < 7
- C.
  x ≤ −3 or x > 7
- D.
  −3 ≤ x < 7
Answer: D. −3 ≤ x < 7
'At least −3' includes −3 (≤), and 'less than 7' excludes 7 (<), so −3 ≤ x < 7. A swaps which endpoint is included; B excludes −3, which 'at least' allows; C describes numbers outside the range rather than between.
Question 3
For which values of x is 2(3 − x) ≥ 4x − 18?
- A.
  x ≤ −2
- B.
  x ≥ 4
- C.
  x ≤ 4.8
- D.
  x ≤ 4
Answer: D. x ≤ 4
6 − 2x ≥ 4x − 18 gives 24 ≥ 6x, so x ≤ 4. B comes from writing −6x ≥ −24 and dividing by −6 without reversing; A moves −18 as −18 instead of +18 (−12 ≥ 6x); C multiplies only the 3 by 2 (6 − x ≥ 4x − 18).
Question 4
A weather station in Kerala recorded 12 mm of rain per day for the first 15 days of a month. For a 30-day month, the total must stay at or below 450 mm. What is the greatest average daily rainfall, in mm, allowed for the last 15 days?
- A.
  9
- B.
  18
- C.
  30
- D.
  270
Answer: B. 18
So far 180 mm has fallen, leaving 270 mm for 15 days, and 270 ÷ 15 = 18 mm per day. Choice A divides the 270 mm over all 30 days; C divides the whole 450 mm limit by 15 days; D is the remaining total, not a daily average.
Question 5
What is the greatest integer value of x that satisfies -4x - 7 > 10?
- A.
  -5
- B.
  -4
- C.
  -3
- D.
  5
Answer: A. -5
Add 7: -4x > 17. Dividing by the negative number -4 reverses the inequality: x < -17/4. The greatest integer less than -17/4 is -5.

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