📖 What Is a Linear Inequality?
A linear inequality is similar to a linear equation, but instead of an equals sign (=), it uses an inequality symbol:
≥
Greater than or equal to
📌 Key Concept
The solution to an inequality is a range of values rather than a single value. For example, x > 3 means any number greater than 3.
⚠️ Important Rule for Inequalities
🚨 CRITICAL RULE
When you multiply or divide both sides by a negative number, you must reverse the inequality sign!
Example:
−3x < 9 → x > −3 (sign reversed!)
⚡ Types of Linear Inequalities
🔹 One-Variable Inequalities
Inequalities with one variable, solved like equations but with special sign rules.
🔹 Two-Variable Inequalities
Inequalities with two variables, graphed as shaded regions on the coordinate plane.
- Dashed line → < or > (not included)
- Solid line → ≤ or ≥ (included)
- Shaded region → all points that satisfy the inequality
📝 Solved Examples
Study these examples carefully. Each shows the step-by-step solution process.
EXAMPLE 1 — One-Variable Inequality
Solve for x: 3x + 7 > 22
Step 1: Subtract 7 from both sides: 3x > 15
Step 2: Divide by 3 (positive, sign stays): x > 5
✅ x > 5
💡 Tip: Any number greater than 5 satisfies the inequality.
EXAMPLE 2 — Sign Reversal
Solve for x: −2x + 4 < 10
Step 1: Subtract 4 from both sides: −2x < 6
Step 2: Divide by −2 (NEGATIVE → REVERSE SIGN!): x > −3
✅ x > −3
💡 Tip: Check: −2(−2) + 4 = 4 + 4 = 8 < 10 ✓ (and −2 > −3)
EXAMPLE 3 — Two-Variable Inequality
Graph the inequality: y ≥ 2x + 1
Step 1: This is already in y = mx + b form: slope = 2, y-intercept = 1
Step 2: Since it's ≥, draw a solid line for y = 2x + 1
Step 3: Shade above the line (y is greater than)
✅ Solid line at y = 2x + 1, shade above
💡 Tip: Test a point: (0,0) → 0 ≥ 1? NO → shade the other side.
EXAMPLE 4 — Testing a Point
Does the point (2, 5) satisfy: 3x + 2y < 12?
Step 1: Substitute x = 2, y = 5: 3(2) + 2(5) = 6 + 10 = 16
Step 2: Is 16 < 12? No, 16 > 12
✅ (2, 5) does NOT satisfy the inequality
💡 Tip: Always test points to verify shaded regions.
EXAMPLE 5 — System of Inequalities
Find the solution region: y > x and y ≤ 4
Step 1: y > x → dashed line y = x, shade above
Step 2: y ≤ 4 → solid line y = 4, shade below
Step 3: The solution is the overlapping region
✅ Overlapping region between y > x and y ≤ 4
💡 Tip: The solution must satisfy BOTH inequalities.
🧪 Practice Questions
Solve each inequality. Click "Show Answer" to see the full solution.
Solve for x: 2x + 5 > 13
A) x > 3
B) x > 4
C) x < 4
D) x > 6
✓ Answer: B
2x + 5 > 13 → 2x > 8 → x > 4
📝 Solution: 2x > 8 → x > 4
Solve for x: 3x − 7 ≤ 14
A) x ≤ 5
B) x ≤ 6
C) x ≤ 7
D) x ≥ 7
✓ Answer: C
3x − 7 ≤ 14 → 3x ≤ 21 → x ≤ 7
📝 Solution: 3x ≤ 21 → x ≤ 7
Solve for x: −4x < 20
A) x < −5
B) x > −5
C) x > −5
D) x < 5
✓ Answer: C
−4x < 20 → Divide by −4 (REVERSE SIGN!) → x > −5
📝 Solution: −4x < 20 → x > −5 (sign reversed!)
Solve for x: 5x + 3 ≥ 2x − 6
A) x ≥ −3
B) x ≤ −3
C) x ≤ 3
D) x ≥ −3
✓ Answer: D
5x + 3 ≥ 2x − 6 → 3x ≥ −9 → x ≥ −3
📝 Solution: 5x − 2x ≥ −6 − 3 → 3x ≥ −9 → x ≥ −3
Solve for x: 6 − 2x < 10
A) x < −2
B) x > −2
C) x < 2
D) x > 2
✓ Answer: B
6 − 2x < 10 → −2x < 4 → Divide by −2 (REVERSE!) → x > −2
📝 Solution: −2x < 4 → x > −2 (sign reversed!)
Which graph represents y > 2x + 1?
A) Solid line, shade below
B) Dashed line, shade below
C) Dashed line, shade above
D) Solid line, shade above
✓ Answer: C
y > 2x + 1 → Dashed line (not included), shade above (y greater than)
📝 Solution: > → dashed line, shade above
Does the point (2, 3) satisfy 4x − y ≥ 5?
✓ Answer: A
Substitute: 4(2) − 3 = 8 − 3 = 5 ≥ 5 ✓ Yes
📝 Solution: 4(2) − 3 = 5 ≥ 5 → Yes
Solve for x: 2x + 3 > 5x − 9
A) x > 4
B) x < −4
C) x < 4
D) x > −4
✓ Answer: C
2x + 3 > 5x − 9 → −3x > −12 → Divide by −3 (REVERSE!) → x < 4
📝 Solution: 2x − 5x > −9 − 3 → −3x > −12 → x < 4
Solve for x: (2x + 1)/3 < 5
A) x < 6
B) x < 7
C) x > 7
D) x < 7
✓ Answer: D
(2x + 1)/3 < 5 → 2x + 1 < 15 → 2x < 14 → x < 7
📝 Solution: 2x + 1 < 15 → 2x < 14 → x < 7
Which point satisfies the system: y > 2x and y ≤ 3?
A) (0, 1)
B) (2, 5)
C) (1, 3)
D) (3, 0)
✓ Answer: C
Check (1, 3): 3 > 2(1) → 3 > 2 ✓ and 3 ≤ 3 ✓
📝 Solution: (1, 3) satisfies both: 3 > 2 and 3 ≤ 3
🎉 WELL DONE!
You've completed the Linear Inequalities lesson. You now know how to solve and graph inequalities, and how to handle the sign reversal rule.