Linear Inequalities — ACT Math Guide

Q: What You Need to Know

• Inequality symbols: (greater than), ≤ (less than or equal), ≥ (greater than or equal) • Solving inequalities follows the same steps as equations, with one crucial exception • When multiplying or dividing both sides by a negative number, flip the inequality symbol • Solutions are r

Linear inequalities ACT questions test your ability to solve and graph expressions with inequality symbols instead of equals signs. These problems involve finding ranges of values that make mathematical statements true, rather than single solutions. You'll encounter approximately 3-4 linear inequality problems on the ACT math section out of 60 questions in 60 minutes. With clear strategies and practice, you can master these problems and boost your ACT math score confidently.

What You Need to Know

Inequality symbols: < (less than), > (greater than), ≤ (less than or equal), ≥ (greater than or equal)

Solving inequalities follows the same steps as equations, with one crucial exception

When multiplying or dividing both sides by a negative number, flip the inequality symbol

Solutions are ranges of values, not single numbers

Graphing shows the solution set on a number line with open or closed circles

📐 KEY FORMULA: ax + b < c becomes x < (c - b)/a (flip symbol if dividing by negative a)

⏱️ ACT TIME TIP: Check your answer by plugging in a test value from your solution range — this catches sign flip errors quickly in the 60-minute time limit.

How to Solve Linear Inequalities on the ACT

Example Question 1 — Easy/Medium Difficulty

If 3x - 7 ≤ 11, which of the following represents all possible values of x?

A) x ≤ 6

B) x ≥ 6

C) x ≤ 18

D) x ≥ 18

E) x ≤ 4

Solution:

Step 1: Add 7 to both sides: 3x - 7 + 7 ≤ 11 + 7, so 3x ≤ 18

Step 2: Divide both sides by 3: x ≤ 6

Step 3: Check with x = 0: 3(0) - 7 = -7, and -7 ≤ 11 ✓

✅Answer: A — The inequality symbol doesn't flip because we divided by positive 3.

Example Question 2 — Hard Difficulty

For which values of x is the inequality -2(3x + 4) > 5x - 6 true?

A) x > -2/11

B) x < -2/11

C) x > 2/11

D) x < 2/11

E) x > -11/2

Solution:

Step 1: Distribute the -2: -6x - 8 > 5x - 6

Step 2: Add 6x to both sides: -8 > 11x - 6

Step 3: Add 6 to both sides: -2 > 11x

Step 4: Divide by 11: -2/11 > x, which means x < -2/11

✅Answer: B — We didn't divide by a negative, so the inequality symbol stays the same.

Common ACT Math Mistakes to Avoid

❌Mistake: Forgetting to flip the inequality when multiplying or dividing by negative numbers

✅Fix: Always check the sign of the number you're dividing by — negative means flip the symbol

❌Mistake: Confusing the direction of inequality symbols on multiple choice answers

✅Fix: Test a value from your solution in the original inequality to verify

❌Mistake: Rushing through distribution and making arithmetic errors

✅Fix: Take 10 extra seconds to double-check your distribution step — it prevents cascading errors

❌Mistake: Mixing up open circles (< or >) with closed circles (≤ or ≥) when graphing

✅Fix: Remember "less than" and "greater than" use open circles; "equal to" gets closed circles

Practice Question — Try It Yourself

Which inequality represents the solution to -4x + 12 ≥ 20?

A) x ≤ -2

B) x ≥ -2

C) x ≤ 2

D) x ≥ 2

E) x ≤ 8

Show Answer

Answer: A — Subtract 12 from both sides: -4x ≥ 8. Divide by -4 and flip the symbol: x ≤ -2.

Key Takeaways for the ACT

Always flip the inequality symbol when multiplying or dividing by negative numbers

Test your final answer with a sample value to catch errors quickly

ACT math linear inequalities often appear in word problems — translate carefully

Remember that solutions are ranges, not single values

Use your calculator freely throughout the ACT math section to check arithmetic