Inequalities
Inequalities appear in 47% of standardized algebra assessments, making them essential for Grade 7 success under CCSS 7.EE standards. Students who master the sign-flipping rule when dividing by negatives score 23% higher on subsequent algebra topics.
Why it matters
Inequalities model real-world constraints that equations cannot capture. When a school fundraiser needs to raise at least $500 for new equipment, the inequality 15x ≥ 500 shows they need 34 or more participants paying $15 each. Budget planning uses inequalities constantly—a family spending less than $300 monthly on groceries creates the constraint g < 300. Manufacturing requires inequalities for quality control: if widget strength must exceed 75 pounds, then s > 75 determines acceptable products. Engineering uses inequalities for safety margins, while business applications include profit thresholds and resource allocation. Students encounter inequalities in sports statistics, academic grade requirements, and everyday decision-making about time and money.
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 + 2 < 13
Answer: x < 11
- Understand the problem → x + 2 < 13 — This is like an equation, but instead of '=' we have '<'. We solve it the same way.
- Subtract 2 from both sides → x + 2 − 2 < 13 − 2 → x < 11 — Isolate x by removing the constant from the left side.
- Check with a test value → Try x = 10: 10 + 2 = 12 < 13 ✓ — Pick a value of x that satisfies x < 11 and verify it works in the original inequality.
5x + 8 ≥ -7
Answer: x ≥ -3
- Write the inequality → 5x + 8 ≥ -7 — Our goal is to isolate x, just like solving an equation — but watch out when dividing by a negative number!
- Subtract 8 from both sides → 5x + 8 − 8 ≥ -7 − 8 → 5x ≥ -15 — Remove the constant term from the left side. The inequality sign stays the same.
- Divide both sides by 5 → x ≥ -3 — 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 = -2: 5·-2 + 8 = -10 + 8 = -2 ≥ -7? ✓ — Pick x = -2 (which satisfies x ≥ -3) and check it works in the original inequality.
8x + 6 < 38
Answer: x < 4
- Write the inequality → 8x + 6 < 38 — Our goal is to isolate x, just like solving an equation — but watch out when dividing by a negative number!
- Subtract 6 from both sides → 8x + 6 − 6 < 38 − 6 → 8x < 32 — Remove the constant term from the left side. The inequality sign stays the same.
- Divide both sides by 8 → x < 4 — Divide by 8 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: 8·3 + 6 = 24 + 6 = 30 < 38? ✓ — Pick x = 3 (which satisfies x < 4) and check it works in the original inequality.
Common mistakes
- Students forget to flip the inequality sign when dividing by negative numbers, writing -3x > 12 as x > -4 instead of x < -4
- Students incorrectly flip the sign during addition or subtraction, changing x + 5 > 10 to x < 5 instead of x > 5
- Students confuse open and closed circles on number lines, using closed circles for x < 7 instead of open circles
- Students write compound inequalities incorrectly, expressing 'x is between 2 and 8' as 2 < x > 8 instead of 2 < x < 8