Math

Linear Inequalities in 1 or 2 Variables

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Inequalities show up constantly on the test, usually hidden inside word problems about budgets, minimum scores, and 'at least'/'at most' situations. The math is almost identical to solving equations — with one sneaky rule that trips up half of test-takers.

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An inequality is like an equation, but instead of an equals sign it uses one of these comparison symbols:

> means greater than
< means less than
≥ means greater than or equal to ("at least")
≤ means less than or equal to ("at most")

So x > 5 means "x is any number bigger than 5" — not just one answer, but a whole range.

Solving a one-variable inequality works exactly like solving an equation: you isolate the variable by adding, subtracting, multiplying, and dividing on both sides. Example: solve 3x - 7 > 14. Add 7 to both sides → 3x > 21. Divide by 3 → x > 7.

BUT here's THE rule you must memorize:

When you multiply or divide both sides by a NEGATIVE number, you FLIP the inequality sign.

For example, -2x ≤ -6. Divide both sides by -2 and flip the ≤ into ≥: x ≥ 3. If you forget to flip, you get the exact wrong half of the answer.

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Solution set for x ≥ 3 — the closed (filled) end means 3 is included.

Translating words into symbols is half the battle on word problems:

"at least" → ≥
"at most" → ≤
"more than" → >
"fewer than / less than" → <
"no more than" → ≤
"minimum" / "needs" → ≥

Translating words to inequality symbols

Phrase	Symbol	Meaning
at least / minimum	≥	this value or more
at most / no more than	≤	this value or less
more than / greater than	>	strictly above
fewer than / less than	<	strictly below

Keyword-to-symbol cheat sheet for word problems.

Two-variable inequalities look like y > 2x + 1. Instead of a single line, the solution is a whole region of the coordinate plane — every (x, y) point that makes the statement true. You graph the boundary line y = 2x + 1, then shade above (for > or ≥) or below (for < or ≤). Use a dashed line for < or > (boundary not included) and a solid line for ≤ or ≥ (boundary included).

To test which side to shade, plug in an easy point like (0,0). If it makes the inequality true, shade that side; if not, shade the other side.

Most test questions are one-variable solve-for-x problems or word problems. Master the flip rule and the translation table and you'll handle the majority of them.

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Quick check

Check your understanding with a question from this topic:

If 3x - 7 > 14, what is the least integer value of x that satisfies the inequality?

Your answer:

Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).

Worked examples

Example 1

If 3x - 7 > 14, what is the least integer value of x that satisfies the inequality?

Example 2

Which of the following values of x satisfies the inequality -2x + 9 ≤ 3?

Example 3

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

Common pitfalls

Forgetting to flip the sign with a negative

Whenever you multiply or divide both sides by a negative number, the inequality sign MUST flip (< becomes >, etc.). Skipping this gives you the wrong half of the number line — a trap the test deliberately includes as an answer choice.

Confusing strict vs. inclusive boundaries

x > 7 does NOT include 7, so the least integer is 8; but x ≥ 7 DOES include 7. Read the symbol carefully — 'least integer' and 'minimum' questions hinge on this distinction.

Mistranslating 'at least' and 'at most'

'At least' means ≥ (you can have more), and 'at most' means ≤ (you can have less). Many students swap these. Anchor on: 'at least' sets a floor, 'at most' sets a ceiling.

Wrong shading direction in two-variable graphs

Don't guess which side to shade. Plug in a test point like (0,0): if it satisfies the inequality, shade that side; if not, shade the opposite side. Also match dashed (strict) vs. solid (inclusive) lines.

Key takeaways

Solve inequalities just like equations — except FLIP the sign when multiplying or dividing by a negative number.
'At least' and 'minimum' mean ≥; 'at most' and 'no more than' mean ≤.
Strict inequalities (< or >) exclude the boundary value; inclusive ones (≤ or ≥) include it.
For two-variable inequalities, graph the boundary line (dashed for strict, solid for inclusive) and test a point to decide which side to shade.
On word problems, define a variable, translate the words into a symbol, then solve.

Watch & learn

Curated Khan Academy walkthroughs on Linear Inequalities in 1 or 2 Variables. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Linear Inequalities in 1 or 2 Variables, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Linear Inequalities in 1 or 2 Variables

You are here

→Linear Equations in 1 Variable →Systems of 2 Linear Equations in 2 Variables →Linear Functions