SAT — mathFoundational5 min read

Solving Linear Inequalities

Master solving linear inequalities for the Digital SAT. Learn when to flip the inequality sign and how to express solutions.

Tutor AI Team2026-02-25T01:17:12.160Z

Linear inequalities are like linear equations, except instead of finding a single value that makes both sides equal, you find a range of values that make one side greater (or less) than the other. The Digital SAT tests your ability to solve inequalities, interpret their solutions, and apply them to real-world contexts.

The solving process is almost identical to solving equations, with one critical difference: you must flip the inequality sign when multiplying or dividing by a negative number.

Core Concepts

Inequality Symbols

Symbol	Meaning	Number Line
$<$	less than	open circle
$>$	greater than	open circle
$\leq$	less than or equal to	closed circle
$\geq$	greater than or equal to	closed circle

Solving One-Step Inequalities

Use the same inverse operations as equations.

$x + 5 > 12$ → $x > 7$

$3x \leq 18$ → $x \leq 6$

The Critical Rule: Flipping the Sign

When you multiply or divide both sides by a negative number, you must reverse the inequality sign.

$-2x > 8$ → divide by $-2$ → $x < -4$ (sign flipped!)

Why? Because multiplying by a negative reverses the order. If $a > b$ , then $-a < -b$ .

Solving Multi-Step Inequalities

Follow the same process as multi-step equations:

Example: Solve $4x - 7 \geq 13$

Add 7: $4x \geq 20$

Divide by 4: $x \geq 5$

Example: Solve $-3x + 9 < 0$

Subtract 9: $-3x < -9$

Divide by $-3$ (flip!): $x > 3$

Representing Solutions on a Number Line

$x > 3$ : open circle at 3, arrow to the right →
$x \leq -2$ : closed circle at $-2$ , arrow to the left ←

Inequalities in Context

Example: A phone plan costs $30/month plus $0.10 per text. If the budget is at most $50, how many texts can be sent?

$30 + 0.10t \leq 50$ → $0.10t \leq 20$ → $t \leq 200$

At most 200 texts.

Strategy Tips

Tip 1: Treat It Like an Equation (Almost)

Solve as you would an equation. The only extra step: flip the sign when multiplying/dividing by a negative.

Tip 2: Check with a Test Value

Pick a number in your solution set and plug it into the original inequality to verify.

Tip 3: Flip the Sign — Don't Forget!

This is the most common error. Any time you multiply or divide by a negative, the direction reverses.

Tip 4: Context Tells You the Direction

"At most" = $\leq$ . "At least" = $\geq$ . "More than" = $>$ . "Fewer than" = $<$ .

Tip 5: Inequalities Can Be Answer Choices

The SAT may give answer choices like $x > 3$ , $x < 3$ , $x \geq 3$ , $x \leq 3$ . Solve carefully to pick the right one.

Worked Example: Example 1

Problem

Solve $5x + 3 > 28$

$5x > 25$ → $x > 5$

Solution

Worked Example: Example 2

Problem

Solve $-6x + 12 \leq 42$

$-6x \leq 30$ → $x \geq -5$ (flipped!)

Solution

Worked Example: Example 3

Problem

Solve $2(x - 4) > 3x + 1$

$2x - 8 > 3x + 1$

$-8 - 1 > 3x - 2x$

$-9 > x$ → $x < -9$

Solution

Worked Example: SAT-Style

Problem

A student needs at least 80 points on average across 4 tests. The first three scores are 72, 85, and 79. What is the minimum score on the fourth test?

$\frac{72 + 85 + 79 + x}{4} \geq 80$

$236 + x \geq 320$

$x \geq 84$

Minimum score: 84.

Solution

Worked Example: Example 5

Problem

Solve $\frac{3 - x}{2} \geq 4$

$3 - x \geq 8$

$-x \geq 5$

$x \leq -5$ (flipped!)

Solution

Practice Problems

Problem 1

Solve $7x - 4 < 31$ .

Problem 2

Solve $-5x + 10 \geq -15$ .

Problem 3

Solve $3(2x + 1) \leq 4x + 13$ .

Problem 4

A parking garage charges $5 plus $2 per hour. You have $19. For how many hours can you park?

Problem 5

Solve $\frac{-2x + 6}{3} > 2$ .

Problem 6

If $-4x + 8 < 2x - 10$ , what is the solution?

Want to check your answers and get step-by-step solutions?

Common Mistakes

Forgetting to flip the inequality when dividing by a negative. This is the #1 error.
Flipping the inequality when it's not needed. Only flip when multiplying/dividing by a negative, not when subtracting.
Confusing $<$ with $\leq$ . "Less than" is strict ( $<$ ); "at most" or "no more than" includes equality ( $\leq$ ).
Wrong direction on the number line. $x > 3$ means numbers to the right of 3, not to the left.
Not checking the answer. Always test a value in your solution range.

Frequently Asked Questions

When exactly do I flip the inequality sign?

Only when you multiply or divide both sides by a negative number. Never when adding or subtracting.

Can I solve inequalities using a calculator?

On Desmos, you can graph both sides as functions and see where one is above or below the other.

What does a solution like $x > 5$ look like on the SAT?

It might be an answer choice written as $x > 5$ , or the SAT might ask you to fill in a number line, or the answer might be "all real numbers greater than 5."

Are there compound inequalities on the SAT?

Yes — questions like "if $2 < x < 8$ , what is the range of $3x + 1$ ?" do appear.

What's the difference between $>$ and $\geq$ on the SAT?

Strict inequality ( $>$ ) excludes the endpoint; $\geq$ includes it. Read the context: "more than" vs. "at least."

Key Takeaways

✓
Solve inequalities like equations, with one key difference.
✓
Flip the inequality sign when multiplying or dividing by a negative.
✓
Check your answer with a test value from the solution set.
✓
Context keywords: "at most" = $\leq$ , "at least" = $\geq$ , "more than" = $>$ , "fewer than" = $<$ .
✓
Number line: open circle for strict inequality, closed circle for inclusive.
✓
Practise flipping the sign until it's automatic.

Tags:

#sat#math#algebra#inequalities

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Core Concepts

Inequality Symbols

Solving One-Step Inequalities

The Critical Rule: Flipping the Sign

Solving Multi-Step Inequalities

Representing Solutions on a Number Line

Inequalities in Context

Strategy Tips

Tip 1: Treat It Like an Equation (Almost)

Tip 2: Check with a Test Value

Tip 3: Flip the Sign — Don't Forget!

Tip 4: Context Tells You the Direction

Tip 5: Inequalities Can Be Answer Choices

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: SAT-Style

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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