QuestionSolve the inequality $|x-7|+12 \leq 25$ and select the correct solution set from the options given.

Studdy Solution

STEP 1

Assumptions1. The inequality given is $|x-7|+12 \leq25$ . The absolute value function $|x-7|$ represents the distance of $x$ from $7$ on the number line3. The inequality $|x-7|+12 \leq25$ means that the distance of $x$ from $7$ , plus $12$ , is less than or equal to $25$

STEP 2

First, we need to isolate the absolute value function on one side of the inequality. We can do this by subtracting $12$ from both sides of the inequality.
$|x-7|+12 -12 \leq25 -12$

STEP 3

implify the inequality.
$|x-7| \leq25 -12$

STEP 4

Calculate the right side of the inequality.
$|x-7| \leq13$

STEP 5

The inequality $|x-7| \leq13$ means that the distance of $x$ from $7$ is less than or equal to $13$ . This can be rewritten as two separate inequalities $x-7 \leq13$ and $x-7 \geq -13$ .

STEP 6

olve the first inequality $x- \leq13$ by adding $$ to both sides.
$x-+ \leq13+$

STEP 7

implify the first inequality.
$x \leq20$

STEP 8

olve the second inequality $x-7 \geq -13$ by adding $7$ to both sides.
$x-7+7 \geq -13+7$

STEP 9

implify the second inequality.
$x \geq -6$

STEP 10

The solution to the inequality $|x-7|+12 \leq25$ is the intersection of the solutions to the two inequalities $x \leq20$ and $x \geq -6$ . This can be written as the interval $[-6,20]$ .
The correct answer is option d. $[-6,20]$ .

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