Part of step 5 of 20·Algebra·Linear Inequalities in 1 or 2 Variables Prev Next

Math

Linear Inequalities

4 min readEasy10-question checkpoint

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.

Where this lesson fits

Before

Linear Equations→

Linear Inequalities

You are here

→Linear Functions →Systems of Equations

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.

Loading…

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.

Loading…

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) → $>$

Loading…

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.

A student needs at least 560 points total on 7 quizzes to earn an A in a course. After 6 quizzes, the student has 475 points. What is the minimum score the student needs on the 7th quiz?

Worked examples

Example 1

Solve for $x$ :

-3x + 7 \geq 19

Loading…

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.

Loading…

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.

Loading…

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.

Loading…

Confusing the boundary

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

Loading…

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 → $>$ .
Loading…
Solutions are ranges, not single values — $x > 3$ means every number greater than 3.
Loading…
Always check by plugging a test value back into the original inequality.

Checkpoint

10 questions on Linear Inequalities in 1 or 2 Variables, from our question bank. Score 70% or better across 10 answers to reach Silver and pass this step.

Start checkpoint · 10 questions

Where this lesson fits

Before

Linear Equations→

Linear Inequalities

You are here

→Linear Functions →Systems of Equations

Worked examples

Common pitfalls

Key takeaways

Further reading

Checkpoint