Answer :
To solve the inequality
[tex]$$
|x-5| + 2 < 20,
$$[/tex]
we follow these steps:
1. Isolate the absolute value:
Subtract 2 from both sides:
[tex]$$
|x-5| + 2 - 2 < 20 - 2 \quad \Longrightarrow \quad |x-5| < 18.
$$[/tex]
2. Remove the absolute value:
The inequality [tex]$|x-5| < 18$[/tex] means that the expression inside the absolute value is between [tex]$-18$[/tex] and [tex]$18$[/tex]. This is written as:
[tex]$$
-18 < x-5 < 18.
$$[/tex]
3. Solve for [tex]$x$[/tex]:
Add 5 to every part of the inequality to solve for [tex]$x$[/tex]:
[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]
[tex]$$
|x-5| + 2 < 20,
$$[/tex]
we follow these steps:
1. Isolate the absolute value:
Subtract 2 from both sides:
[tex]$$
|x-5| + 2 - 2 < 20 - 2 \quad \Longrightarrow \quad |x-5| < 18.
$$[/tex]
2. Remove the absolute value:
The inequality [tex]$|x-5| < 18$[/tex] means that the expression inside the absolute value is between [tex]$-18$[/tex] and [tex]$18$[/tex]. This is written as:
[tex]$$
-18 < x-5 < 18.
$$[/tex]
3. Solve for [tex]$x$[/tex]:
Add 5 to every part of the inequality to solve for [tex]$x$[/tex]:
[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]
