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Maths 9 – Inequalities and systems of inequalities

📘 Notes

Inequalities and Systems of Inequalities

In this topic, you solve linear inequalities, double inequalities, product inequalities, quadratic inequalities, and systems of inequalities.

For a system of inequalities, the final answer is the intersection of the solutions. For quadratic inequalities, you can use the graphical method or the interval method.

1. Linear inequalities

Solve them almost like equations.

The most important rule:

If you multiply or divide by a negative number, the inequality sign changes direction.

Example:

-2x + 7 < -5

-2x < -12

x > 6

2. Double inequalities

A double inequality means two inequalities at the same time.

Example: -1 ≤ -2x + 3 < 5

Solve step by step. If you divide by a negative number, reverse both signs.

3. Systems of inequalities

First solve each inequality separately.

Then find the intersection.

Intersection means the values that satisfy both inequalities.

Example: if one inequality gives x ≥ -6 and the other gives x > 6, then the intersection is x > 6.

4. Product inequalities

For inequalities such as (2x - 6)(x + 1) < 0, first find the zeros of each factor.

2x - 6 = 0, so x = 3

x + 1 = 0, so x = -1

Then mark these points on a number line and use the interval method.

Find where the product is positive or negative, then choose the interval that matches the sign in the question.

5. Quadratic inequalities

For quadratic inequalities, you can use:

  • Graphical method: sketch the parabola and see where it is above or below the x-axis.
  • Interval method: find the roots, mark them on a number line, and determine the sign in each interval.

Example: x2 - 25 < 0

x2 - 25 = (x - 5)(x + 5)

Roots: x = -5 and x = 5

Since the parabola opens upwards, the expression is negative between the roots. So the answer is -5 < x < 5.

6. Standard methods

Method 1: Linear inequality

  1. Move all x-terms to one side and numbers to the other.
  2. Simplify.
  3. If dividing by a negative number, reverse the sign.

Method 2: System of inequalities

  1. Solve each inequality separately.
  2. Write each answer clearly.
  3. Take the intersection.

Method 3: Product inequality

  1. Find the zeros of each factor.
  2. Mark them on a number line.
  3. Use the interval method to find the sign in each interval.
  4. Choose the correct interval.

Method 4: Quadratic inequality

  1. Factorise if possible.
  2. Find the roots.
  3. Use a graph or the interval method.
  4. Include endpoints for ≤ or ≥, but not for < or >.

7. Worked examples

Example 1: System of inequalities

Solve the system:

2x - 2 ≥ x - 8

-2x + 7 < -5

Solution

First inequality: x ≥ -6

Second inequality: x > 6

Intersection: x > 6

Example 2: Product inequality

Solve (2x - 6)(x + 1) < 0.

Solution

Zeros: x = 3 and x = -1

The product is negative between the roots, so -1 < x < 3.

Example 3: Double inequality

Solve -1 ≤ -2x + 3 < 5.

Solution

-1 ≤ -2x + 3 < 5

-4 ≤ -2x < 2

Divide by -2 and reverse the signs:

2 ≥ x > -1

So the answer is -1 < x ≤ 2.

Example 4: Quadratic inequality

Solve x2 + 10x ≥ 0.

Solution

x2 + 10x = x(x + 10)

Roots: x = 0 and x = -10

The expression is non-negative outside the roots, so x ≤ -10 or x ≥ 0.

Example 5: Quadratic inequality

Solve x2 - 25 < 0.

Solution

x2 - 25 = (x - 5)(x + 5)

Roots: x = -5 and x = 5

The expression is negative between the roots, so -5 < x < 5.

8. Endpoints

  • Use a closed endpoint for ≤ or ≥.
  • Use an open endpoint for < or >.
  • In a system, always take the overlap only.

9. Common mistakes

  • Forgetting to reverse the sign when dividing by a negative number.
  • For a system, writing the union instead of the intersection.
  • For product inequalities, not using intervals.
  • For quadratic inequalities, choosing the wrong interval.
  • Including endpoints when the sign is strict.