2010 AMC 10A Problem 11

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.

Problem 11 in other years