Inequalities
An inequality compares two expressions using symbols like <, >, ≤, or ≥ instead of an equals sign. Solving inequalities follows the same steps as solving equations, with one crucial exception: multiplying or dividing both sides by a negative number flips the inequality sign. For example, -2x > 6 becomes x < -3 after dividing by -2.
Why it matters
Inequalities model countless real-world constraints and comparisons. A business needs revenue > $50,000 to break even, or a bridge must support weight ≤ 80 tons. Manufacturing requires temperatures between 150°F and 200°F, expressed as 150 ≤ T ≤ 200. Sports statistics use inequalities to define qualifying standards, like a runner needing a time < 12.5 seconds. Budget planning involves inequalities: total expenses < available funds. In advanced mathematics, inequalities define domains of functions, optimization problems in calculus, and solution regions in linear programming. The CCSS 7.EE standards emphasize solving one- and two-step inequalities, building foundations for algebra concepts in high school where inequalities appear in systems, absolute value problems, and rational functions.
How to solve inequalities
Inequalities
- Solve like an equation (same operations on both sides).
- If you multiply or divide by a negative, FLIP the sign.
- Graph on a number line (open circle for < >, closed for ≤ ≥).
Example: -2x > 6 → x < -3 (sign flipped).
Worked examples
x + 5 > 15
Answer: x > 10
- Understand the problem → x + 5 > 15 — This is like an equation, but instead of '=' we have '>'. We solve it the same way.
- Subtract 5 from both sides → x + 5 − 5 > 15 − 5 → x > 10 — Isolate x by removing the constant from the left side.
- Check with a test value → Try x = 11: 11 + 5 = 16 > 15 ✓ — Pick a value of x that satisfies x > 10 and verify it works in the original inequality.
4x + 1 < 37
Answer: x < 9
- Write the inequality → 4x + 1 < 37 — Our goal is to isolate x, just like solving an equation — but watch out when dividing by a negative number!
- Subtract 1 from both sides → 4x + 1 − 1 < 37 − 1 → 4x < 36 — Remove the constant term from the left side. The inequality sign stays the same.
- Divide both sides by 4 → x < 9 — Divide by 4 to isolate x. The inequality sign stays the same since we're dividing by a positive number.
- Verify with a test value → Try x = 8: 4·8 + 1 = 32 + 1 = 33 < 37? ✓ — Pick x = 8 (which satisfies x < 9) and check it works in the original inequality.
3x + 10 > 7
Answer: x > -1
- Write the inequality → 3x + 10 > 7 — Our goal is to isolate x, just like solving an equation — but watch out when dividing by a negative number!
- Subtract 10 from both sides → 3x + 10 − 10 > 7 − 10 → 3x > -3 — Remove the constant term from the left side. The inequality sign stays the same.
- Divide both sides by 3 → x > -1 — Divide by 3 to isolate x. The inequality sign stays the same since we're dividing by a positive number.
- Verify with a test value → Try x = 0: 3·0 + 10 = 0 + 10 = 10 > 7? ✓ — Pick x = 0 (which satisfies x > -1) and check it works in the original inequality.
Common mistakes
- When dividing -3x < 15 by -3, writing x < -5 instead of x > -5, forgetting to flip the inequality sign when dividing by a negative number.
- Solving 2x + 4 ≥ 10 and writing x ≥ 7 instead of x ≥ 3, making arithmetic errors during the isolation process.
- Graphing x > 5 with a closed circle instead of an open circle, confusing the symbols > and ≥ on number lines.