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# Solving Inequalities. How To Solve A Linear Inequality. Math Help.

Solving an inequality can be done in a similar way to an equation – that is applying inverses to both side of the inequality to find x. The only exception is when you multiply or divide by a negative number. If this happens then you need to turn around the inequality sign. You can see this from the inequality 2<5 (2 is less than 5), if you divide both sides by -1 you get -2<-5 which isn’t correct (as -2 is not less than -5), but if you turn the inequality sign around you get -2 >-5 which is now correct (as -2 is more than -5).

Let’s take a look at some questions that involve solving inequalities.

**Example 1**

Solve the inequality 3x – 2 < 4

3x – 2 < 4 (Add 2 to both sides)

3x < 6 (Divide both sides by 3)

x < 2

**Example 2**

Solve the inequality 9x + 1 > 64

9x + 1 > 64 (Subtract 1 from both sides)

9x > 63 (Divide both sides by 9)

x> 7

**Example 3**

Solve the inequality -3x - 4 > -19.

-3x - 4 > -19 (Add 4 to both sides)

-3x > -15 (Divide both side by -3)

x < 5

Make sure that the inequality sign is turned around as in the last example you had to divide both sides of the inequality by a negative number.

**Example 4**

Solve the inequality (x/-3) - 8 ≤ 4

(x/-3) - 8 ≤ 4 (First add 8 to both sides of the inequality).

(x/-3) ≤ 12 (Next multiply both sides by -3)

x ≥ -36

Again, the inequality sign has to be turned around as both sides of the inequality were multiplied by a negative number in the last step.

**Example 5**

Solve the inequality 9x + 2 > 4x + 32

5x + 2 > 32 (First subtract 4x from both sides of the inequality)

5x > 30 (Subtract 2 from both sides of the inequality)

x > 6 (Divide both sides of the inequality by 5)

**Example 6**

Solve the inequality 3x + 2 > 7(x+ 10)

3x + 2 > 7(x+ 10) (First expand the bracket on the right hand side)

3x + 2 > 7x + 70 (Next subtract 7x from both sides of the inequality)

-4x + 2 > 70 (Next subtract 2 from both sides of the inequality)

-4x > 68 (Divide both sides of the inequality by -4)

x < 16 (Make sure you turn the inequality sign around again as you had to divide by a negative number)

Basically, solving inequalities can be done in the same way as solving an equation. Just remember to switch the inequality around if the inequality was divided or multiplied by a negative number.