We are asked to find and graph the solution set for all possible values of x in the given inequality.
|2x-4| ≤ 11
To do this, we will create a compound inequality by removing the absolute value. In this case, the solution set is any number less than or equal to $11$ away from the midpoint in the positive direction and any number less than or equal to $11$ away from the midpoint in the negative direction.
\begin{aligned}
\textbf{Absolute Value Inequality: }& \ \ \phantom{\N 11\leq} |2x-4| \leq 11 \\
\textbf{Compound Inequality: }& \ \ \N 11\leq \ 2x-4 \ \leq 11
\end{aligned}
We can split this compound inequality into two cases, one where $2x-4$ is greater than or equal to $\N11$ and one where $2x-4$ is less than or equal to $11.$
\begin{aligned}
\N 11\leq2x-4\quad \text{and} \quad 2x-4\leq 11
\end{aligned}
Let's isolate $x$ in both of these cases before graphing the solution set.
This inequality tells us that all values greater than or equal to $\N 3.5$ will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the overlap of the solution sets. Let's recombine our cases back into one compound inequality.
\begin{aligned}
\textbf{First Solution Set:}&\quad \ \phantom{\N 3.5\leq} x \leq 7.5\\
\textbf{Second Solution Set:}&\quad \ \N 3.5 \leq x \\
\textbf{Intersecting Solution Set:}& \quad \ \N 3.5\leq x \leq 7.5
\end{aligned}
Graph
The graph of this inequality includes all values from $\N 3.5$ to $7.5,$ inclusive. We show this by using closed circles on the endpoints.
