Linear Inequalities
A linear inequality is similar to a linear equation, but instead of an equals sign (=), it uses inequality symbols to show that one side is greater than or less than the other.
Visualizing Inequalities
Use the interactive graph below to see how the shaded region changes for the inequality y > 2x + 3. You can zoom in, out, and drag the graph to explore the solution set.
Inequality Symbols
- < : Less than
- > : Greater than
- ≤ : Less than or equal to
- ≥ : Greater than or equal to
The Golden Rule of Inequalities
⚠️ CRITICAL RULE:
When you multiply or divide both sides of an inequality by a negative number, you must FLIP the inequality sign.
Example Problem
Solve for x: -3x + 4 > 10
Step 1: Subtract 4 from both sides:
-3x > 6
Step 2: Divide by -3 and FLIP the sign:
x < -2
Practice Quiz
1. Solve for y: 2y – 5 ≤ 7
Show Answer
Answer: y ≤ 6. (Add 5 to both sides: 2y ≤ 12, then divide by 2.)
2. Solve for x: x + 8 > 3
Show Answer
Answer: x > -5. (Subtract 8 from both sides: 3 – 8 = -5.)
3. Solve for m: -4m < 20
Show Answer
Answer: m > -5. (Divide by -4 and FLIP the sign.)
4. Solve for a: 5a + 2 ≥ 17
Show Answer
Answer: a ≥ 3. (Subtract 2: 5a ≥ 15, then divide by 5.)
5. Solve for p: p/3 – 1 ≤ 4
Show Answer
Answer: p ≤ 15. (Add 1: p/3 ≤ 5, then multiply by 3.)
🔥 Tricky & Hard Challenges
6. Solve for x: 7 – 2(x + 4) ≥ 15
Show Step-by-Step Solution
Step 1: Distribute the -2: 7 – 2x – 8 ≥ 15
Step 2: Combine like terms: -2x – 1 ≥ 15
Step 3: Add 1: -2x ≥ 16
Step 4: Divide by -2 and FLIP THE SIGN: x ≤ -8
7. Solve for z: (3z – 9) / -3 > z + 2
Show Step-by-Step Solution
Step 1: Multiply both sides by -3 and FLIP THE SIGN: 3z – 9 < -3(z + 2)
Step 2: Distribute: 3z – 9 < -3z - 6
Step 3: Add 3z to both sides: 6z – 9 < -6
Step 4: Add 9: 6z < 3
Step 5: Divide by 6: z < 0.5