Solve the inequality: $|2x-3|\le 5$

Key concept: An absolute value inequality of the form $|A|\le b$ becomes a double inequality, which can then be solved step by step.

Step by step

We solve

$|2x-3|\le 5$

Step 1: Rewrite as a compound inequality

For $|A|\le 5$,

$-5\le A\le 5$

So,

$-5\le 2x-3\le 5$

Step 2: Add 3 to all three parts

$-2\le 2x\le 8$

Step 3: Divide by 2

$-1\le x\le 4$

Answer

$\boxed{[-1,4]}$

So the correct choice is D.

Pitfall alert

A frequent mistake is turning $|2x-3|\le 5$ into only one inequality, such as $2x-3\le 5$. You must include both sides: $-5\le 2x-3\le 5$. Another common error is forgetting to divide every part of the inequality by 2 at the end.

Try different conditions

If the sign were $|2x-3|<5$, the solution would be $-1<x<4$, so the interval would be $(-1,4)$. If it were $|2x-3|\ge 5$, the answer would be two separate intervals: $x\le -1$ or $x\ge 4$.

Further reading

absolute value, compound inequality, interval notation