Math

Linear Inequalities

5 min readEasy5-question drill

Linear inequalities work just like linear equations — same distribute, combine, isolate moves — with one critical twist: when you multiply or divide by a negative, the inequality sign flips.

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Linear Equations→

Linear Inequalities

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A linear inequality uses $<$ , $\leq$ , $>$ , or $\geq$ instead of $=$ . The solution isn't a single value — it's a range of values. For example, $x > 3$ means every number greater than 3.

Most solving moves work exactly like equations:

Add or subtract any quantity to both sides — sign stays the same.
Multiply or divide by a positive number — sign stays the same.
Multiply or divide by a negative number — flip the inequality sign.

That last rule is the only thing that makes inequalities different from equations.

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When does the sign flip? When does it stay?

Operation	Example	Sign flips?
Add / subtract anything	$x - 3 < 5$ → $x < 8$	No
Multiply / divide by POSITIVE	$2x > 8$ → $x > 4$	No
Multiply / divide by NEGATIVE	$-2x > 8$ → $x < -4$	Yes — flip

Why does the sign flip? If $5 > 3$ , multiply both by $-1$ : you get $-5$ on the left, $-3$ on the right. Is $-5 > -3$ ? No — $-5$ is less than $-3$ . The order reversed. Hence the flip.

Compound inequalities — sometimes you'll see $-3 < 2x + 1 \leq 7$ . Treat it as two inequalities and do the same operation to all three parts:

Subtract 1: $-4 < 2x \leq 6$ .
Divide by 2: $-2 < x \leq 3$ .

The answer is the range $-2 < x \leq 3$ — every value between them, including 3 but not -2.

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Open dot = strict inequality, value not included. Closed dot = ≤ or ≥, boundary value included.

Common SAT tasks:

Solve for $x$ and pick the choice with the right range.
Pick the value that is (or is not) a solution. Plug each option into the original.
Word problems: translate at least → $\geq$ , no more than → $\leq$ , fewer than → $<$ .

Word translations to memorize:

At least, no less than, minimum → $\geq$
At most, no more than, maximum → $\leq$
Fewer than, less than (strict) → $<$
More than, greater than (strict) → $>$

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Quick check

Solve as you would a linear equation — but watch the moment you multiply or divide by a negative. Flip the inequality sign every time.

What is the least integer value of x such that 6x - 3 ≥ 11?

Worked examples

Example 1

Solve for $x$ : $-3x + 7 \geq 19$

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Example 2

A theater has 240 seats. The owner wants ticket revenue from a single show to be at least $3,600. If tickets cost $15 each, what is the minimum number of tickets that must be sold?

Common pitfalls

Forgetting to flip the sign

Multiply or divide by a negative → flip $<$ to $>$ (and vice versa). This is the #1 inequality mistake on the SAT. Practice flagging it the moment you see a negative coefficient.

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Flipping the sign for addition / subtraction

The flip rule applies ONLY to multiplication and division by a negative. Adding or subtracting (even a negative number) does NOT flip the sign. " $x + 3 < 7$ → subtract 3 → $x < 4$ " — no flip needed.

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Mis-translating word problems

"At least 50" means $\geq 50$ , not $> 50$ . "No more than 50" means $\leq 50$ . "Fewer than 50" means $< 50$ . Read the problem carefully — strict vs. non-strict matters.

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Confusing the boundary

$x \leq 4$ INCLUDES $x = 4$ . $x < 4$ does NOT. The boundary inclusion matters when picking specific values or graphing.

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Key takeaways

Linear inequalities solve like linear equations — with one rule: flip the sign when multiplying or dividing by a negative.
Adding or subtracting (even negative numbers) does NOT flip the sign.
Translate words carefully: at least → $\geq$ , at most → $\leq$ , fewer than → $<$ , more than → $>$ .
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Solutions are ranges, not single values — $x > 3$ means every number greater than 3.
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Always check by plugging a test value back into the original inequality.

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Try it yourself

5 practice questions on Linear Inequalities, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Before

Linear Equations→

Linear Inequalities

You are here

→Linear Functions →Systems of Equations