Question

which of the following represents the solution to the inequality 2|5 - 2x| - 3 ≤ 15?
(-∞,-2)∪(7,∞)
(-∞,1.5)∪(7.5,∞)
-2,7
1.5,7.5

Explanation:

Step1: Isolate the absolute - value expression

Add 3 to both sides of the inequality $2|5 - 2x|-3\leq15$.
$2|5 - 2x|\leq15 + 3$, so $2|5 - 2x|\leq18$. Then divide both sides by 2, we get $|5 - 2x|\leq9$.

Step2: Rewrite the absolute - value inequality as a compound inequality

If $|a|\leq b$ ($b\geq0$), then $-b\leq a\leq b$. So, $-9\leq5 - 2x\leq9$.

Step3: Solve the left - hand side of the compound inequality

Solve $-9\leq5 - 2x$. Subtract 5 from both sides: $-9-5\leq - 2x$, so $-14\leq - 2x$. Divide both sides by - 2 and reverse the inequality sign (since dividing by a negative number), we have $x\leq7$.

Step4: Solve the right - hand side of the compound inequality

Solve $5 - 2x\leq9$. Subtract 5 from both sides: $-2x\leq9 - 5$, so $-2x\leq4$. Divide both sides by - 2 and reverse the inequality sign, we get $x\geq - 2$.

Answer:

$[-2,7]$