Begin with the inequality

6x − 7 ≤ 5

Adding 7 to both sides gives

6x ≤ 12

Dividing both sides by 6 gives

x ≤ 2

This is represented on the number line below.

Begin with the inequality

4(x − 2) > 20

First divide both sides of the inequality by 4 to give

x − 2 > 5

Then adding 2 to both sides of the inequality gives x > 7.

Begin with the inequality

5 − 6x ≥ −19

In this case, note that the inequality contains a '−6x' term. The first step here is to add 6x to both sides, giving

5 ≥ −19 + 6x

Now 19 can be added to both sides to give

24 ≥ 6x

Then dividing both sides by 6 gives

4 ≥ x

or

x ≤ 4

Begin with the inequality

−10 < 6x + 2 ≤ 32

The same operation must be performed on each part of the inequality. The first step is to subtract 2, which gives

−12 < 6x ≤ 30

Then dividing by 6 gives

−2 < x ≤ 5

The result can then be represented on a number line as shown below.

An alternative approach is to consider the inequality as two separate inequalities:

(1)   −10 < 6x + 2     and     (2)   6x + 2 ≤ 32

These can be solved as shown below.

Both inequalities can then be displayed as shown below.

Where the two lines overlap gives the solution as