Review question

Can we solve this inequality involving two modulus functions? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9896

Solution

Find the set of real values of $x$ for which $\big |x+3\big |>2\big |x-3\big |.$

The values of $x$ for which the inequality is satisfied are the values of $x$ for which the graph of $\big |x+3\big |$ lies above the graph of $2\big |x-3\big |$, so it is first helpful to sketch the graphs of $y=\big |x+3\big |$ and $y=2\big |x-3\big |$ on the same axes:

From this sketch, we see that the region where the inequality is satisfied will be of the form $a < x < b$.

We see that $a$ is the $x$-coordinate of the intersection of the lines $y=x+3$ and $y=-2(x-3)$, so $a$ is the solution to $x+3=-2x+6$, which gives $a=1$.

The value $b$ is the x-coordinate of the intersection of the lines $y=x+3$ and $y=2(x-3)$, so it is the solution of $x+3=2x-6$, which gives $b=9$.

Therefore the region where the inequality is satisfied is $1<x<9$.