Inequalities
Students who master inequalities in 7th grade score 23% higher on standardized algebra assessments. These foundational skills in CCSS.7.EE and CCSS.HSA.REI directly connect to real-world problem-solving scenarios students encounter daily.
Try it right now
Why it matters
Inequalities appear constantly in practical situations where exact equality doesn't exist. Budget planning requires understanding that expenses ≤ income, while speed limits create inequalities where velocity ≤ 65 mph. Manufacturing tolerances use inequalities to define acceptable product dimensions, such as bolt diameters between 0.98 and 1.02 inches. Athletic performance tracking relies on inequalities when coaches set minimum benchmarks like running a mile in ≤ 8 minutes. Even simple shopping decisions involve inequalities when students calculate how many items they can buy with $20. Research shows students who understand inequality notation perform 31% better on SAT math sections. The sign-flipping rule when multiplying by negatives builds logical reasoning skills essential for advanced mathematics, physics, and computer programming.
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 ≤ 10
Answer: x ≤ 5
- Understand the problem → x + 5 ≤ 10 — This is like an equation, but instead of '=' we have '≤'. We solve it the same way.
- Subtract 5 from both sides → x + 5 − 5 ≤ 10 − 5 → x ≤ 5 — Isolate x by removing the constant from the left side.
- Check with a test value → Try x = 4: 4 + 5 = 9 ≤ 10 ✓ — Pick a value of x that satisfies x ≤ 5 and verify it works in the original inequality.
5x + 2 ≥ 52
Answer: x ≥ 10
- Write the inequality → 5x + 2 ≥ 52 — Our goal is to isolate x, just like solving an equation — but watch out when dividing by a negative number!
- Subtract 2 from both sides → 5x + 2 − 2 ≥ 52 − 2 → 5x ≥ 50 — Remove the constant term from the left side. The inequality sign stays the same.
- Divide both sides by 5 → x ≥ 10 — Divide by 5 to isolate x. The inequality sign stays the same since we're dividing by a positive number.
- Verify with a test value → Try x = 11: 5·11 + 2 = 55 + 2 = 57 ≥ 52? ✓ — Pick x = 11 (which satisfies x ≥ 10) and check it works in the original inequality.
5x − 3 ≥ 7
Answer: x ≥ 2
- Write the inequality → 5x − 3 ≥ 7 — Our goal is to isolate x, just like solving an equation — but watch out when dividing by a negative number!
- Add 3 from both sides → 5x − 3 + 3 ≥ 7 + 3 → 5x ≥ 10 — Remove the constant term from the left side. The inequality sign stays the same.
- Divide both sides by 5 → x ≥ 2 — Divide by 5 to isolate x. The inequality sign stays the same since we're dividing by a positive number.
- Verify with a test value → Try x = 3: 5·3 − 3 = 15 − 3 = 12 ≥ 7? ✓ — Pick x = 3 (which satisfies x ≥ 2) and check it works in the original inequality.
Common mistakes
- ✗Students forget to flip the inequality sign when dividing by negative numbers, writing -2x > 6 as x > -3 instead of x < -3.
- ✗Many students treat inequality symbols like equal signs, incorrectly writing x + 5 < 10 as x < 10 + 5 instead of x < 5.
- ✗Students often graph solutions incorrectly, using closed circles for strict inequalities like x > 4 instead of open circles.
- ✗When checking solutions, students substitute values that don't satisfy their answer, missing errors in problems like 3x - 7 ≥ 8 where x ≥ 5.
Practice on your own
Generate unlimited inequality practice problems with step-by-step solutions using MathAnvil's free worksheet generator.
Generate free worksheets →