We have already seen in example 1.2b that the solution to $\displaystyle \frac{x^2+2x+3}{2x^2+x-6}<0$ is $-2 < x < \frac{3}{2}$.

Follow-up question: solve $\displaystyle \frac{x^2 + 2|x| + 3}{2x^2 + |x| - 6} < 0$.

Notice that $x$ in the original expression has been replaced with $|x|$ (do you know why the $x^2$ term remains $x^2$ and does not need to be changed to $|x|^2$?).

For such follow-up questions, it is not necessary to repeat the earlier steps again. Instead, we can use the previous solutions $-2 < x <\frac{3}{2}$ and replace $x$ with $|x|$ to get $-2 < |x| < \frac{3}{2}$. We then solve this inequality (graphs can come in handy. You may also want to refer to example 1.1 ). This gives the solution $-\frac{3}{2} < x < \frac{3}{2}$.

Other common substitutions

The following examples showcase various common substitutions, starting from $\displaystyle \frac{x^2+2x+3}{2x^2+x-6}<0$. Are you able to identify them?
Hover or click on each question to reveal the substitution and resulting answer.