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].
[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].
