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

$\mathrm{(A)}\ 6 \qquad \mathrm{(B)}\ 10 \qquad \mathrm{(C)}\ 20 \qquad \mathrm{(D)}\ 30 \qquad \mathrm{(E)}\ 40$

I looked at the solutions and saw that $x$ has to be separated and then the two sides subtracted are equal to 10. I do not understand what is meant by the "interval of solutions" and why $x$ is separated.

1 Answers 1

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Any solution of an inequation is of the form $p \le x \le q $. So the length of the interval is then $ q -p$. Here if we keep x in the middle then $ p = (a-3)/2 , q = (b-3)/2 $. Thus then the length in this case $(b-a)/2 = 10$. Thus the answer is 20.