Linear Inequalities in One Variable — SAT Math Guide

Q: What You Need to Know

• Inequality symbols: (greater than), ≤ (less than or equal), ≥ (greater than or equal) • Solve inequalities like equations, but flip the inequality sign when multiplying or dividing by a negative number • Solutions are ranges of values, not single numbers • Graph solutions on numbe

Linear inequalities in one variable SAT problems test your ability to solve and graph inequalities with symbols like <, >, ≤, and ≥. These problems involve finding ranges of values that make an inequality true, rather than just one specific answer. You'll encounter approximately 2-3 linear inequality questions on the Digital SAT math section. With the right approach, these questions become straightforward points you can confidently earn.

What You Need to Know

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

Solve inequalities like equations, but flip the inequality sign when multiplying or dividing by a negative number

Solutions are ranges of values, not single numbers

Graph solutions on number lines using open circles (< or >) or closed circles (≤ or ≥)

Check your answer by substituting a value from your solution range back into the original inequality

📐 KEY FORMULA: When solving ax + b < c, isolate x to get x < (c - b)/a

💡 PRO TIP: Always flip the inequality sign when multiplying or dividing both sides by a negative number — this is the most common SAT trap!

How to Solve Linear Inequalities in One Variable on the SAT

Example Question 1 — Medium Difficulty

If 3x + 7 > 22, which of the following could be a value of x?

A) 4

B) 5

C) 6

D) 7

Solution:

Step 1: Subtract 7 from both sides: 3x > 15

Step 2: Divide both sides by 3: x > 5

Step 3: Check which answer choice satisfies x > 5

✅Answer: C — Only 6 and 7 are greater than 5, but since we need just one possible value, C) 6 works perfectly.

Example Question 2 — Hard Difficulty

For what values of x is the inequality -2(x - 4) ≥ 3x - 2 true?

A) x ≤ 2

B) x ≥ 2

C) x ≤ 10

D) x ≥ 10

Solution:

Step 1: Distribute the -2: -2x + 8 ≥ 3x - 2

Step 2: Add 2x to both sides: 8 ≥ 5x - 2

Step 3: Add 2 to both sides: 10 ≥ 5x

Step 4: Divide by 5: 2 ≥ x, which means x ≤ 2

✅Answer: A — The solution x ≤ 2 means x can be 2 or any number less than 2.

Common SAT Math Mistakes to Avoid

❌Mistake: Forgetting to flip the inequality sign when dividing by a negative number

✅Fix: Always remember that dividing or multiplying by negative numbers reverses the inequality direction

❌Mistake: Confusing "at least" and "at most" language in word problems

✅Fix: "At least" means ≥ (greater than or equal), "at most" means ≤ (less than or equal)

❌Mistake: Mixing up open and closed circles on number line graphs

✅Fix: Use closed circles for ≤ or ≥, open circles for < or >

❌Mistake: Not checking your answer by substituting back into the original inequality

✅Fix: Pick a number from your solution range and verify it works in the original problem

Practice Question — Try It Yourself

Solve the inequality: -4x + 12 < 8

A) x > 1

B) x < 1

C) x > -1

D) x < -1

Show Answer

Answer: A — Subtract 12: -4x < -4, then divide by -4 (flip the sign): x > 1

Key Takeaways for the SAT

Treat inequalities like equations, but remember to flip the sign when multiplying or dividing by negatives

Your SAT math score improves when you double-check by substituting a test value back into the original inequality

Pay attention to "at least" and "at most" language in Digital SAT word problems — they signal ≥ and ≤ respectively

Practice translating between inequality notation, number line graphs, and interval notation

Most College Board linear inequality questions test the sign-flipping rule, so master this concept