Inequalities – Grade 9 Maths

Linear Inequalities

Solve and represent on a number line

Integer Solutions

List whole number values satisfying an inequality

Double Inequalities

Compound inequalities: a < x < b

Quadratic Inequalities

x² > a or x² < a and their solutions

Regions on a Graph

Shade regions satisfying multiple inequalities

1. Solving Linear Inequalities

Solving a linear inequality is similar to solving an equation — you perform the same operations on both sides. The critical rule is: when you multiply or divide by a negative number, reverse the inequality sign.

Solve: 3x − 4 > 8
Add 4: 3x > 12
Divide by 3: x > 4

Inequality symbols:
< strictly less than | ≤ less than or equal to
> strictly greater than | ≥ greater than or equal to

Remember: multiplying or dividing both sides by a negative number flips the inequality sign. E.g., −2x < 6 → x > −3.

2. Integer Solutions

When asked to list integer solutions, solve the inequality first, then list all whole numbers (integers) in that range.

Solve: 2 < x ≤ 7, list integers.
Integers satisfying: 3, 4, 5, 6, 7

For "largest integer satisfying x < 4.6": list the integers less than 4.6 → …, 2, 3, 4. The largest is 4.
For "smallest integer satisfying x > −2.3": the smallest integer greater than −2.3 is −2.

Use a number line to visualise the solution set before listing integers.

3. Double Inequalities

A double inequality restricts x between two bounds simultaneously: a < x < b or a ≤ x ≤ b.

Solve: −1 < 2x + 3 < 11
Subtract 3 throughout: −4 < 2x < 8
Divide by 2: −2 < x < 4

Perform the same operation on all three parts simultaneously. The solution represents a range of values for x.

Always keep the inequality signs pointing in the same direction throughout the solution.

4. Quadratic Inequalities

To solve a quadratic inequality such as x² − 4 > 0 or x² < 9, find the critical values (roots) and then determine which region satisfies the inequality.

x² < 9: roots at x = ±3. The parabola is below zero between the roots: −3 < x < 3
x² > 9: solution is x < −3 or x > 3

Method:
1. Rearrange to get f(x) > 0 or f(x) < 0.
2. Find roots by factorising or using the quadratic formula.
3. Sketch the parabola to determine the solution region.

For a positive leading coefficient: the parabola is above zero outside the roots and below zero between them.

5. Regions on a Graph and Set Notation

Inequalities in two variables (e.g., y > 2x + 1) define half-planes on a coordinate grid. The solution set is a region.

Graphing an inequality:
1. Draw the boundary line (solid for ≤ or ≥; dashed for < or >).
2. Test a point (e.g., the origin) to determine which side satisfies the inequality.
3. Shade the required region.

Set notation:
{x : x > 3} means "the set of all x such that x is greater than 3".
{x : −2 ≤ x < 5} is the set of real numbers from −2 to 5, including −2 but not 5.

When solving linear programming problems, the feasible region is the intersection of multiple inequality regions.

Example 1 — Solving a Linear Inequality

Solve 5x − 3 ≥ 12 and represent on a number line.

Add 3: 5x ≥ 15

Divide by 5: x ≥ 3

Number line: Solid dot at 3, arrow pointing right.

Example 2 — Reversing the Inequality

Solve −3x + 2 > 11

Subtract 2: −3x > 9

Divide by −3 (reverse sign!): x < −3

Check: Try x = −4: −3(−4) + 2 = 14 > 11 ✓

Example 3 — Double Inequality

Solve −5 < 3x + 1 ≤ 10 and list all integer solutions.

Subtract 1: −6 < 3x ≤ 9

Divide by 3: −2 < x ≤ 3

Integer solutions: −1, 0, 1, 2, 3

Example 4 — Quadratic Inequality

Solve x² − 25 < 0.

Factor: (x − 5)(x + 5) < 0

Roots: x = 5 and x = −5

Parabola is negative between roots: −5 < x < 5

Example 5 — Integer Solutions from Compound Inequality

Find the integers satisfying 2 ≤ 3x − 1 < 14.

Add 1: 3 ≤ 3x < 15

Divide by 3: 1 ≤ x < 5

Integer solutions: 1, 2, 3, 4

Inequality Grapher

Enter a linear inequality in x. The number line will highlight the solution set.

Coefficient of x: Sign: Constant (right side): Subtract from x side first: (i.e. ax+b sign c)

Exercise 1 — Solve Linear Inequality (give boundary value)

1. Solve 2x − 3 < 3. Give the boundary value of x.

2. Solve 3x + 1 ≤ 16. Give the boundary value of x.

3. Solve x + 4 > 2. Give the boundary value of x.

4. Solve 4x − 4 ≥ 12. Give the boundary value of x.

5. Solve 5x − 3 < 32. Give the boundary value of x.

6. Solve x + 3 < 2. Give the boundary value of x.

7. Solve 2x − 2 > 2. Give the boundary value of x.

8. Solve 3x + 3 ≤ 21. Give the boundary value of x.

9. Solve −x − 1 < 2. Give the boundary value of x (i.e. x > ?)

10. Solve 4x − 4 < 28. Give the boundary value of x.

Exercise 2 — Largest Integer Satisfying the Inequality

1. Find the largest integer satisfying 2x − 3 < 6.

2. Find the largest integer satisfying x + 1 ≤ 7.

3. Find the largest integer satisfying 3x − 1 < 8.

4. Find the largest integer satisfying 2x ≤ 16.

5. Find the largest integer satisfying x − 2 < 2.

6. Find the largest integer satisfying 3x − 5 ≤ 25.

7. Find the largest integer satisfying x + 5 < 7.

8. Find the largest integer satisfying 2x − 3 < 12.

9. Find the largest integer satisfying 4x − 3 ≤ 17.

10. Find the largest integer satisfying x + 4 ≤ 13.

Exercise 3 — Double Inequality (give lower bound)

1. Solve −7 < 2x − 1 < 5. Give the lower bound of x.

2. Solve −11 < 2x − 1 < 7. Give the lower bound of x.

3. Solve −3 < 2x − 1 < 9. Give the lower bound of x.

4. Solve −9 < 2x − 1 < 7. Give the lower bound of x.

5. Solve −5 < 2x − 1 < 11. Give the lower bound of x.

6. Solve −13 < 2x − 1 < 9. Give the lower bound of x.

7. Solve −15 < 2x − 1 < 11. Give the lower bound of x.

8. Solve −1 < 2x − 1 < 13. Give the lower bound of x.

9. Solve −17 < 2x − 1 < 7. Give the lower bound of x.

10. Solve −19 < 2x − 1 < 5. Give the lower bound of x.

Exercise 4 — Double Inequality (give upper bound)

1. Solve −7 < 2x − 1 < 5. Give the upper bound of x.

2. Solve −11 < 2x − 1 < 7. Give the upper bound of x.

3. Solve −3 < 2x − 1 < 9. Give the upper bound of x.

4. Solve −9 < 2x − 1 < 7. Give the upper bound of x.

5. Solve −5 < 2x − 1 < 11. Give the upper bound of x.

6. Solve −13 < 2x − 1 < 9. Give the upper bound of x.

7. Solve −15 < 2x − 1 < 11. Give the upper bound of x.

8. Solve −1 < 2x − 1 < 13. Give the upper bound of x.

9. Solve −17 < 2x − 1 < 7. Give the upper bound of x.

10. Solve −19 < 2x − 1 < 5. Give the upper bound of x.

Exercise 5 — Quadratic Inequality (give positive boundary)

1. Solve x² < 9. Give the positive boundary value.

2. Solve x² > 25. Give the positive boundary value.

3. Solve x² ≤ 16. Give the positive boundary value.

4. Solve x² < 4. Give the positive boundary value.

5. Solve x² > 36. Give the positive boundary value.

6. Solve x² ≥ 49. Give the positive boundary value.

7. Solve x² < 1. Give the positive boundary value.

8. Solve x² > 64. Give the positive boundary value.

9. Solve x² ≤ 81. Give the positive boundary value.

10. Solve x² > 100. Give the positive boundary value.

Practice — 20 Questions

1. Solve 2x − 3 < 3. Boundary value?

2. Solve 3x + 1 ≤ 16. Boundary value?

3. Solve x + 4 > 2. Boundary value?

4. Solve 4x − 4 ≥ 12. Boundary value?

5. Solve 5x − 3 < 32. Boundary value?

6. Solve x + 3 < 2. Boundary value?

7. Solve 2x − 2 > 2. Boundary value?

8. Solve 3x + 3 ≤ 21. Boundary value?

9. Solve −x − 1 < 2. Boundary value (x > ?).

10. Solve 4x − 4 < 28. Boundary value?

11. Largest integer: 2x − 3 < 6.

12. Largest integer: x + 1 ≤ 7.

13. Largest integer: 3x − 1 < 8.

14. Largest integer: 2x ≤ 16.

15. Largest integer: x − 2 < 2.

16. Largest integer: 3x − 5 ≤ 25.

17. Largest integer: x + 5 < 7.

18. Largest integer: 2x − 3 < 12.

19. Largest integer: 4x − 3 ≤ 17.

20. Largest integer: x + 4 ≤ 13.

Challenge — 8 Questions

1. Solve −7 < 2x − 1 < 5. Lower bound?

2. Solve −11 < 2x − 1 < 7. Lower bound?

3. Solve −3 < 2x − 1 < 9. Lower bound?

4. Solve −9 < 2x − 1 < 7. Lower bound?

5. Solve −5 < 2x − 1 < 11. Lower bound?

6. Solve −13 < 2x − 1 < 9. Lower bound?

7. Solve −15 < 2x − 1 < 11. Lower bound?

8. Solve −1 < 2x − 1 < 13. Lower bound?