Question

How do you solve the inequality `abs(2x+1)<=6-x` and write your answer in interval notation?

Answer 1

`-7 \leq x \leq 5/3`

Explanation:

Since `|a|\leq b \Leftrightarrow -b \leq a \leq b`, the inequality `|2x+1|\leq 6-x`becomes

`-(6-x) \leq 2x+1 \leq 6-x`

`\Leftrightarrow x-6 \leq 2x+1 \leq 6-x`

`\Leftrightarrow x-7 \leq 2x \leq 5-x`

Solve the inequality on the left:
`x-7 \leq 2x \Leftrightarrow -7 \leq x`

Solve the inequality on the right:
`2x \leq 5-x \Leftrightarrow 3x \leq 5 \Leftrightarrow x \leq 5/3`

Combine the two intervals: `-7 \leq x \leq 5/3`