As with regular linear equations, we aim to isolate the variable by subtracting any constants when dividing by the x coefficient. The only major difference is that when we divide or multiply by a negative number, the inequality sign is reversed.

For example, the following inequality is true:

−3<−2

When we multiply both sides by −2, the inequality sign must reverse:

6>4

a)

To put x on its own, we need to add 3 to both sides of the inequality.

x−3<8x<8+3x<11.

b)

In this example we find x by dividing both sides by the coefficient of x, 5.

5x<10x<2.

c)

3x−4<73x<7+4Add 8 to get x on its own.x<7+43 Divide by 3.x<113.

d)

In this example, take the constants to one side, and keep the x term on the other. Divide through by the negative x-coefficient to find an inequality for x. Notice that where you divide (or multiply) an equality by a negative value, the inequality sign is reversed.

−3x−4<7−3x<7+4Add 4 to both sides.x>−7+43 Divide by −3. The inequality is reversed.x>−113.

e)

In this example, separate the constants and the x-term, then divide by the x-coefficient to find an inequality for x.

20x−13<20−18x38x<33x<3338.

f)

In this example, separate the x-term from all other terms and remember to reverse the inequality when dividing by −14.

7a−17x>20+b−3x−14x>20+b−7ax<−20−b+7a14.

g)

In this example, a simple way to solve for x is to divide by −3 before rearranging the rest of the equation by subtracting g from both sides.

−3(x+g)>6h−9x+g<−6h3+3x<−6h3+3−gx<3−2h−g.