Linear Inequalities in 1 or 2 Variables
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.
Watch & learn
Short walkthroughs on Linear Inequalities in 1 or 2 Variables — ours, plus curated picks. Watch one if a written explanation isn't clicking, or after to reinforce.
Description of the figure:
- A number line from negative 2 to 8, with a tick at every integer.
- The segment from 3 to 8 is shaded, with filled circles (endpoints included).
- Marked points at 0, 3, and 7.
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" →
≥
| 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.
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?
Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).
Worked examples
If 3x - 7 > 14, what is the least integer value of x that satisfies the inequality?
Which of the following values of x satisfies the inequality -2x + 9 ≤ 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
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.
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.
'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.
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.
Skill check
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.