Solving inequalities is exactly like solving normal equations, with one important exception. To understand that exception, look at the situations listed below for the following inequality:

5 < 10
Notice how it is currently a true statement.


Multiply both sides of the inequality by a positive number
Watch what happens when you multiply both sides by 5:

The inequality is still true!


Multiply both sides of the inequality by a negative number
But watch what happens when you multiply both sides by –5:

Notice how the inequality is no longer true!


Dividing both sides of the inequality by a positive number
Watch what happens when you divide both sides by 5:

The inequality is true!


Dividing both sides of the inequality by a negative number
But watch what happens when you divide both sides by –5:

Now the inequality is not true!

The Rule When Solving an Inequality: You always reverse the inequality when you multiply or divide by a negative number. Any other time, you keep the inequality the same!

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Example 1

This first example is a one-step problem.


Start by dividing both sides by –5.


Dividing –5 and multiplying –5 cancel each other out. You are then left with:


Since we divided both sides by a negative number, we reverse the inequality.

This tells us that any value of x that is greater than –9 will make the original inequality true!

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Example 2

This second example is a two-step problem.


Start by subtracting 5 from both sides.


We are left with the following (notice that the inequality sign has not changed!):


Next we need to divide both sides by –2.


We are left with the following:


Since we divided both sides by a negative number, we reverse the inequality.

This tells us that any value of x that is greater than –20 will make the original inequality true!