4.06 Solving linear inequalities

When we solve equations, we can apply the same operation to both sides and the equation remains true. Consider the following equation: x+7=12

We can subtract 7 from both sides of the equation in order to find the value of x. This is because both sides of the equation are identical, so what we do to one side, we should do to the other side.

When working with inequalities, we can follow exactly the same procedure, but with an inequality sign in place of the equals sign. But there are two operations that we need to be careful of.

Consider the inequality 9<15.

If we add or subtract both sides by any number, say 3, we can see that the resulting inequality remains true.

\text{Adding }3 \text{ to }9 \text{ and } 15 \\ 9+3 < 15 +3\\12<18

The inequality stays the same and it is true that 12 is less than 18.

\text{Subtracting }3 \text{ from }9 \text{ and } 15 \\ 9-3 < 15 -3\\6<12

Once again the inequality stays the same and it is true that 6 is less than 12.

Adding a negative would have the same effect as subtracting, so we can also add and subtract negative numbers without changing the inequality.

What happens if we multiply or divide both sides?

Let's look at what happens when we multiply by a positive number:

The inequality stays the same, and it is true that 27 is less than 45.

Now let's look at what happens when we divide by a positive number:

The inequality stays the same, and it is true that 3 is less than 5.

But now let's look what happens when we multiply by a negative number:

\begin{aligned} 9& &< &&&15 &&\text{Given} \\ 9 \times \left(-3\right)& &⬚ &&&15 \times \left(-3\right) &&\text{Multiply both sides by} -3 \\ -27& &⬚ &&&-45 &&\text{Evaluate} \end{aligned}

What inequality sign goes in the boxes on lines two and three? Can we just keep the original inequality sign?

If we look at the last line, we can see that -27 is actually greater than -45, as it further to the right on a number line. So we have to reverse the inequality sign whenever we multiply by a negative number. This gives us the following:

So we can see that -27 is greater than -45.

To divide by a negative we will need to use the same trick:

So once again, the inequality switches from < to >.

Now that we have seen what happens when we perform addition, subtraction, multiplication and division, we can use this knowledge to solve inequalities.

Before jumping in algebraically, it can be helpful to consider some possible solutions and non-solutions. Then we can look at an algebraic strategy.

Examples

Much as with  solving equations  from written descriptions, there are certain key words or phrases to look out for. When it comes to inequalities, we now have a few extra key words and phrases to represent the different inequality symbols.

Phrases that can be used to represent an inequality symbol:

• > greater than, more than
• \geq greater than or equal to, at least, no less than
• < less than
• \leq less than or equal to, at most, no more than

Examples