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
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.
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)
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.
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.