ACT Math: Inequalities
Inequalities extend equation-solving to situations with a range of possible answers, and they're tested through solving, graphing, and word problems alike. This free, complete lesson covers solving and graphing linear inequalities, including the crucial sign-flip rule, compound inequalities, absolute value inequalities, and real-world word problems that translate into inequalities. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
1. Solving and Graphing Linear Inequalities
Solving an inequality works just like solving an equation, with one critical exception: multiplying or dividing both sides by a negative number reverses the inequality symbol.
x ≥ −5: closed circle, shaded to the right
Worked Example: Solve −3x + 7 ≤ 22.
−3x ≤ 15
Divide both sides by −3, and flip the inequality symbol:
x ≥ −5
1. Solve: x − 5 > 12
Explanation: Add 5 to both sides: x > 17.
2. Solve: 3x ≤ 21
Explanation: Divide both sides by 3 (positive, so the symbol doesn't flip): x ≤ 7.
3. Solve: −4x < 20
Explanation: Divide both sides by −4 (negative, so flip the symbol): x > −5.
4. Solve: 2x + 5 ≥ 17
Explanation: Subtract 5: 2x ≥ 12. Divide by 2: x ≥ 6.
5. Solve: −3x + 7 ≤ 22
Explanation: This is the worked example above: subtract 7, then divide by −3, flipping the symbol, giving x ≥ −5.
2. Compound Inequalities
A compound inequality like a < expression ≤ b combines two conditions at once. Solve it by performing the same operation to all three parts simultaneously.
Worked Example: Solve −3 < 2x + 1 ≤ 9.
−4 < 2x ≤ 8 (subtract 1 from all three parts)
−2 < x ≤ 4 (divide all three parts by 2)
1. Solve: −6 ≤ 3x < 12
Explanation: Divide all three parts by 3: −2 ≤ x < 4.
2. Solve: 5 < x + 2 < 9
Explanation: Subtract 2 from all three parts: 3 < x < 7.
3. Solve: −2 ≤ x − 32 ≤ 5
Explanation: Multiply all three parts by 2: −4 ≤ x − 3 ≤ 10. Add 3 to all three parts: −1 ≤ x ≤ 13.
4. Which compound inequality represents all real numbers greater than −2 AND less than or equal to 6?
Explanation: "Greater than −2" excludes −2, using <. "Less than or equal to 6" includes 6, using ≤. Together: −2 < x ≤ 6.
5. Solve: −12 < 4x ≤ 24
Explanation: Divide all three parts by 4: −3 < x ≤ 6.
3. Absolute Value Inequalities
An absolute value inequality splits into two cases. For |expression| < k (a "less than" inequality), the result is a single compound inequality: −k < expression < k. For |expression| > k (a "greater than" inequality), the result is two separate, non-overlapping inequalities: expression > k, or expression < −k.
Worked Example: Solve |x − 4| < 6.
−6 < x − 4 < 6
−2 < x < 10
1. Solve: |x + 2| < 7
Explanation: −7 < x + 2 < 7. Subtract 2 from all parts: −9 < x < 5.
2. Solve: |3x − 6| ≤ 12
Explanation: −12 ≤ 3x − 6 ≤ 12. Add 6: −6 ≤ 3x ≤ 18. Divide by 3: −2 ≤ x ≤ 6.
3. Solve: |x − 5| > 3
Explanation: Since this is a "greater than," split into two separate cases: x − 5 > 3 gives x > 8, and x − 5 < −3 gives x < 2.
4. Solve: |2x + 1| ≥ 9
Explanation: 2x + 1 ≥ 9 gives x ≥ 4. 2x + 1 ≤ −9 gives x ≤ −5.
5. Which inequality describes all values of x within 4 units of 10?
Explanation: "Within 4 units of 10" means the distance between x and 10 is less than 4, so |x − 10| < 4.
4. Word Problems and Applications
Inequality word problems often involve phrases like "at least" (≥), "at most" (≤), "no more than" (≤), "exceeds" (>), or "within" (an absolute value inequality). Translate the phrase carefully before setting up the inequality.
1. A parking garage charges $5 for the first hour and $3 for each additional hour. If Sam wants to pay no more than $23 total, what is the maximum number of additional hours he can park?
Explanation: Let h = additional hours: 5 + 3h ≤ 23. Subtract 5: 3h ≤ 18. Divide by 3: h ≤ 6. The maximum is 6 additional hours.
2. A number tripled and then increased by 4 is at least 25. What is the smallest possible integer value of the number?
Explanation: Let x = the number: 3x + 4 ≥ 25. Subtract 4: 3x ≥ 21. Divide by 3: x ≥ 7. The smallest integer satisfying this is 7.
3. The temperature must stay between 65°F and 75°F, inclusive, for a chemical reaction. Write this as a compound inequality, where T is the temperature.
Explanation: "Between 65 and 75, inclusive" means both endpoints are included: 65 ≤ T ≤ 75.
4. A student's test average must be within 5 points of 85 to pass. Write this as an absolute value inequality, where x is the test average.
Explanation: "Within 5 points of 85" means the distance between x and 85 is at most 5: |x − 85| ≤ 5.
Common Mistakes to Avoid
- Forgetting to flip the inequality symbol when multiplying or dividing by a negative number. This is the single most common inequality error, always double-check the sign of what you're multiplying or dividing by.
- Applying an operation to only part of a compound inequality. Whatever you do to one part of a < expression < b, you must do to all three parts, the left side, the middle, and the right side.
- Treating |expression| > k the same way as |expression| < k. A "less than" absolute value inequality produces one connected range (an "and" situation). A "greater than" absolute value inequality produces two separate, non-overlapping rays (an "or" situation). Mixing these up is a very common error.
- Misreading inclusive versus exclusive language. "At least" and "at most" include the boundary value (≥ or ≤); "more than" and "less than" alone do not (> or <).
- Using an open circle when the boundary value is included, or vice versa, on a number line graph. A closed (filled) circle means the endpoint is included (≤ or ≥); an open circle means it isn't (< or >).
Frequently Asked Questions
How can I remember when to flip the inequality sign?
The rule applies only when multiplying or dividing by a negative number, never when adding or subtracting. A quick check: test a simple case like −2 < 3; if you divide both sides by −1 without flipping, you'd get 2 < −3, which is false, confirming the symbol must flip.
What's the difference between a compound inequality and an absolute value inequality?
They're closely related: |expression| < k is actually a shortcut way of writing the compound inequality −k < expression < k. The absolute value notation is just a more compact way to express the same "within a certain distance" idea.
Why do "greater than" absolute value inequalities split into "or" statements instead of "and"?
|expression| > k means the expression is more than k units away from 0 in either direction, positive or negative, so it could be greater than k OR less than −k. These two possibilities don't overlap, which is why they're joined with "or" rather than existing as one continuous range.
Where can I practice more problems like these?
The Inequalities quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.