6.07 Solve inequalities

Consider the inequality 9 < 15. If we add any number on both sides, like 3, we can see that the resulting inequality remains true:

Let's add 3 to 9 and 15:

Now try subtracting 3 from 9 and 15:

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

The inequality remains true because we are maintaining balance on either side of the inequality symbol, like we do with equations.

Addition property of inequality: Adding the same number to each side of an inequality produces an equivalent inequality. Example:

\begin{aligned}&\text{If } &x-2 < 7 \\ &\text{Then } &x-2+2 < 7+2\end{aligned}

Subtraction property of inequality: Subtracting the same number to each side of an inequality produces an equivalent inequality. Example:

\begin{aligned}&\text{If } &x+5 > 7 \\ &\text{Then } &x+5-5 > 7-5\end{aligned}

Let's apply our knowledge of inverses and the addition and subtraction properties of equality to solve some inequalities.

Examples