Question

The length of the interval of solutions of the inequality $a \le 2x + 3 \le b$ is $10.$ What is $b - a?$

$6$

$10$

$15$

$20$

$30$

Solve the inequality for the two endpoints of $x$

The interval length is $\dfrac{b-3}{2}-\dfrac{a-3}{2}$

Splitting the inequality into two of them and solving gives us $a \le 2x + 3$ $x \ge \dfrac{a - 3}{2}$ and $2x + 3 \le b$ $x \le \dfrac{b - 3}{2}.$

The range of the solutions is then $\dfrac{b - 3}{2} - \dfrac{a - 3}{2} = 10,$ which then simplifying gives us $(b - 3) - (a - 3) = 20$ $b - a = 20.$

Thus, D is the correct answer.