## Inequalities

This page covers solving inequalities after studying this section, you will be able to:

- solve inequalities with one variable
- solve inequalities with two variables

**Inequalities: Quadratic Expressions**

^{2}+ 4x + 1 < –2

^{2}+ 4x + 1 = –2

^{2}+ bx + c = 0. All you have to do is bring the –2 to the left hand side.

^{2}+ 4x + 3 = 0.

**Inequalities in one variable**

≤ means 'less than or equal to'

≥ means 'greater than or equal to'

Inequalities can be shown on number lines.

The 1st inequality covers the whole numbers -1, 0, 1. The 2nd inequality only covers 3.

The rules for manipulating inequalities are like those for equations except that multiplying or dividing each side by a negative number changes the direction of the inequality sign.

Examples:

a) Solve the inequality

4(x-2) >x + 10

4x – 8 > x + 10 (Expand the bracket)

4x> x + 18 (Add 8 to both sides)

3x > 18 (Subtract x from both sides)

x> 6 (Divide both sides by 3)

b) Solve

1/3(4 – 4a) ≤ 8

4-4a ≤ 24 (Multiply both sides by 3)

-4a ≤ 20 (Subtract 4 from both sides)

-a ≤ 5 (Divide both sides by 4)

a ≥ -5 (Multiply both sides by -1)

c) Find all the possible integer values of n that satisfy 3n + 1 ≤ 27 < 5n – 6.

Splitting up the inequalities:

3n + 1 ≤ 27

2n ≤ 26, n ≤ 8.66666 so n = 8, 7, 6, 5, …

27 < 5n – 6

33 < 5n, n > 6.6, so n = 7, 8, 9, …

Integers that satisfy both are 7 and 8.

**NOTE:
When working with a double inequalities start by solving the two inequalities. An integer is any whole number, positive negative or zero.**

**When solving inequalities do not forget that multiplying or dividing by a negative number reverses the inequality sign:**

**−x > 3, becomes x < −3 (multiplying by −1).**

**Inequalities in two variables**

For an inequality in 2 variables: 2x - y > 1.

Below is a graph for 2x-y = 1. The formula can be rearanged as y = 2x -1 and results in a straight line.

**NOTE:Remember an equation in the form y = 2x – 1 has gradient 2 and y-intercept (where it crosses the y-axis) of –1, ****in other words y = (gradient) x + (y -intercept)**

The below graph shows the line of 2x-y=1 while the area underneath the line covers values of 2x-y>1 and the area above covers 2x-y<1.

Several inequalities can be shown simultaneously.

The shaded region satisfies y ≤ x, x + y ≤, and y ≥ 1.