College

What is the solution to [tex]|x-5|+2 < 20[/tex]?

A. [tex]-7 < x < 15[/tex]
B. [tex]-13 < x < 23[/tex]
C. [tex]x < -7[/tex] or [tex]x > 15[/tex]
D. [tex]x < -13[/tex] or [tex]x > 23[/tex]

Answer :

We start with the inequality

[tex]$$|x-5| + 2 < 20.$$[/tex]

Step 1. Subtract 2 from both sides:
[tex]$$
|x-5| + 2 - 2 < 20 - 2 \quad \Rightarrow \quad |x-5| < 18.
$$[/tex]

Step 2. Interpret the absolute value inequality. The inequality

[tex]$$|x-5| < 18$$[/tex]

means that the expression [tex]$x-5$[/tex] is within 18 units of 0. This can be written as a compound inequality:

[tex]$$
-18 < x-5 < 18.
$$[/tex]

Step 3. Solve for [tex]$x$[/tex] by adding 5 to each part of the inequality:

[tex]$$
-18 + 5 < x-5 + 5 < 18 + 5,
$$[/tex]

which simplifies to

[tex]$$
-13 < x < 23.
$$[/tex]

Thus, the solution to the inequality is

[tex]$$-13 < x < 23.$$[/tex]

Therefore, the correct answer is [tex]$-13 < x < 23$[/tex].