Introduction to Inequalities
Understand inequality symbols, solve one- and two-step inequalities, graph solutions on a number line, and learn why multiplying by a negative flips the symbol.
Real-world constraints are rarely equalities — they're ranges. You need a credit score above 700. A drug dosage must be between 5 mg and 20 mg. Your budget must stay below $500. Inequalities model these ranges, and solving them tells you the full set of values that satisfy a condition — not just one answer, but a whole region of valid solutions.
- Read and write inequalities using <, >, ≤, ≥, and ≠ symbols
- Solve one- and two-step inequalities using inverse operations
- Graph solution sets on a number line (open vs. closed circles)
- Apply the critical flip rule: multiplying or dividing by a negative reverses the inequality
Inequalities are solved exactly like equations — with one critical exception:
When you multiply or divide both sides by a negative number, you must reverse the inequality symbol.
\[ -2x < 6 \xrightarrow{\div(-2)} x > -3 \]Why? On a number line, multiplying by −1 reflects all numbers — flipping their order. So < becomes >.
Number Line — Solution of x > 3
Open circle at 3 (not included), arrow pointing right (all values greater than 3).
Solve and graph: \(x + 4 > 9\)
- Subtract 4 from both sides: \(x > 5\)
- Graph: open circle at 5, arrow pointing right (all x greater than 5).
Any number greater than 5 satisfies the inequality. Example: \(6 + 4 = 10 > 9\) ✓
Solve: \(-3x + 2 \leq 11\)
- Subtract 2 from both sides: \(-3x \leq 9\)
- Divide by −3 (flip the symbol!): \(x \geq -3\)
- Check with x = 0: \(-3(0) + 2 = 2 \leq 11\) ✓
- Check boundary: \(-3(-3) + 2 = 11 \leq 11\) ✓ (closed circle at −3)
You have $80 to spend. You've already spent $32 on supplies. T-shirts cost $12 each. How many shirts can you buy?
- Write the inequality: \(12s + 32 \leq 80\)
- Subtract 32: \(12s \leq 48\)
- Divide by 12: \(s \leq 4\)
You can buy at most 4 shirts (0, 1, 2, 3, or 4 are all valid).
- \(x - 7 \geq 3\)
- \(-4x < 20\)
- A rider must be at least 48 inches tall. Write and interpret an inequality.
▶ Show Answers
- \(x \geq 10\) — closed circle at 10, arrow right.
- Divide by −4, flip: \(x > -5\) — open circle at −5, arrow right.
- \(h \geq 48\) — closed circle at 48; any height 48 inches or more is valid.
- Forgetting to flip when dividing by a negative: \(-2x < 8\) → \(x > -4\), NOT \(x < -4\). This is the #1 inequality error.
- Open vs. closed circle confusion: < and > use open circles (endpoint excluded); ≤ and ≥ use closed circles (endpoint included).
- Checking only one value: Test a value from inside the solution set AND check the boundary to confirm both are handled correctly.
- Solve inequalities exactly like equations — except for the flip rule.
- Flip the symbol when multiplying or dividing by any negative number.
- Graph on a number line: open circle for strict (<, >), closed for inclusive (≤, ≥).
- Solution sets are ranges — infinitely many values typically satisfy an inequality.
Risk analysts set thresholds using inequalities: a loan is approved if debt-to-income ratio ≤ 0.43, a portfolio is acceptable if standard deviation < 12%, a project is viable if NPV > 0. Understanding inequalities lets you interpret every "minimum requirement," "maximum limit," or "acceptable range" in any professional field.
Calculator Connection
The Linear Inequality Solver solves and graphs one- and two-step inequalities, clearly showing when the flip rule applies. For compound inequalities (like \(-3 < x \leq 7\)), use the Compound Inequality Solver.
Try it with the Calculator
Apply what you've learned with these tools.
Introduction to Inequalities - Quiz
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