How to Solve Inequalities
Inequalities use symbols like <, >, ≤, and ≥ instead of =. They're solved almost like equations — with one critical rule: when you multiply or divide by a negative number, flip the inequality sign.
Solving like equations
For addition and subtraction, treat inequalities exactly like equations. Add or subtract the same amount from both sides; the inequality stays the same.
Example: x − 5 > 12
x > 17
Answer: x > 17 — any number greater than 17 works.
The flip rule
When you multiply or divide both sides by a negative number, reverse the inequality. < becomes >, ≤ becomes ≥, and vice versa.
Example: −2x < 10
Divide both sides by −2. Since we're dividing by a negative, flip < to >:
x > −5
Answer: x > −5
Why? Try x = 0: −2(0) = 0 < 10 ✓. Try x = −10: −2(−10) = 20, which is NOT < 10. So x must be greater than −5.
Graphing solutions
On a number line: use an open circle (○) for < or >, and a closed circle (●) for ≤ or ≥. Shade in the direction of the solution. For x > 3, open circle at 3, shade right.
Compound inequalities
For −3 < x < 5, x is between −3 and 5. Solve each part separately, or treat as one: add 3 to all parts of −3 < x − 3 < 2 to get 0 < x < 5.
Common mistakes
Forgetting to flip the sign: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. −2x < 6 becomes x > −3 (not x < −3).
Open vs. closed circles: Use an open circle for < and > (the value is not included). Use a closed/filled circle for ≤ and ≥ (the value is included).
More examples
Example: Solve 3x + 2 > 11
3x > 9
x > 3
Example: Solve −4x ≤ 20
x ≥ −5 (flip the sign because we divided by −4)
Practice problems
1. Solve: 2x − 5 < 9
Show answer
2. Solve: −3x + 1 ≥ 10
Show answer
3. Solve: 5 − x > 2
Show answer
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