Question

Solve the inequality algebraically. Express your answer using set notation or interval notation. l x-8l greater than or equal to 4. Rewrite the inequality without the absolute values.

Answer 1

To rewrite the inequality:

`\lvert{x-8}\rvert\ge4`

we need to remember that:

`\lvert{x}\rvert\ge a\text{ is equivalent to }x\ge a\text{ or }x\leq-a`

Then in this case we have:

`\begin{gathered} \lvert{x-8}\rvert\ge4 \\ \text{ Is equivalent to:} \\ x-8\ge4\text{ or }x-8\leq-4 \end{gathered}`

Therefore, we can rewrite the inequality as:

`x-8\leq-4\text{ or }x-8\ge4`

Once we have it written in this form we can solve it:

`\begin{gathered} x-8\leq-4\text{ or }x-8\ge4 \\ x\leq-4+8\text{ or }x\ge4+8 \\ x\leq4\text{ or }x\ge12 \end{gathered}`

Therefore, the solution set of the inequality is:

`(-\infty,4\rbrack\cup\lbrack12,\infty)`