1. GCSE Revision 101
Maths
Inequalities
© Daniel Holloway

2. The Basics
3 > -6 is an inequality. It is also a true statement, so 3 > -6 is a true inequality. But does the inequality remain true when you do the following to it?

3. The Basics
add 4 to both sides: > > -2 is still true subtract 2 from both sides: 3 – 2 > -6 – 2 1 > -8 is still true multiply both sides by 2: 3 x 2 > -6 x 2 6 > -12 is still true divide both sides by 3: 3 ÷ 3 > -6 ÷ 3 1 > -2 is still true multiply both sides by -5: 3 x -5 > -6 x > 30 is not true divide both sides by -3: 3 ÷ -3 > -6 ÷ > 2 is not true

4. The Basics This means that to solve an inequality, you can:
add the same quantity to both sides
subtract the same quantity from both sides
multiply both sides by the same positive quantity
divide both sides by the same positive quantity
But you cannot:
multiply both sides by a negative quantity
divide both sides by a negative quantity

5. Solving Simple Inequalities
 Worked Example
a) Solve 3x + 4 > 22 b) Solve 6 – 5x ≥ 3x + 2
a subtract 4 from both sides 3x + 4 – 4 > 22 – x > 18
divide both sides by x ÷ 3 > 18 ÷ x > 6
b add 5x to both sides – 5x + 5x ≥ 3x + 5x ≥ 8x + 2
subtract 2 from both sides 6 – 2 ≥ 8x + 2 – ≥ 8x
divide each side by ÷ 8 ≥ 8x ÷ ½ ≥ x
Another way of writing ½ ≥ x is x ≤ ½

6. Test Yourself
Solve these inequalities: 1) 3x – 5 ≥ 4 5) -2d ≥ -3 2) 5x + 2 > 4 6) -5c < 20 3) 7x + 2 < 5x + 4 7) 5 – ¼f > 12 4) 8 – 3a ≤ 5 8) 2 – ½g ≥ 1

7. Three-sided Inequalities
We have to treat inequalities with three sides as two separate two-sided ones
 Worked Example
Solve 7 ≤ 3x – 2 < 9
Treat this as two inequalities:
7 ≤ 3x – 2 3x – 2 < 9
9 ≤ 3x x < 11
3 ≤ x x < 32/3
We write the answer as: 3 ≤ x < 32/3

8. Regions R Inequalities can be plotted on a graph using regions (R)y x 5
The shaded region here represents the inequality. This region satisfies the inequality:
x > 3
The dotted line means it is “more than but not including” or > R

9. Regions R The shaded region in this grid shows the inequality: y ≤ 6
The filled in line means it is “below and including” or ≤ y x 5 R

10. Regions
We can use multiple regions on the same graph to find points which satisfy several inequalities at the same time
For example, on the following grid, if we wanted to find the region which satisfied the three inequalities:
y < 7
x > 1
y ≥ x

11. Regions y < 7 x > 1 y ≥ x
As you can see, all three inequalities have been drawn on the grid, and this time, those areas which do not fit are shaded (or crossed out) in order to leave the last region
y < 7
x > 1
y ≥ x y 5 x 5