Question $|x+7|-3=8$

Studdy Solution

STEP 1

1. The equation $|x+7|-3=8$ involves an absolute value.
2. We will need to isolate the absolute value term and solve for $x$ considering both cases of the absolute value.

STEP 2

1. Isolate the absolute value expression $|x+7|$ .
2. Solve the equation for both cases of the absolute value.
3. Verify the solutions.

STEP 3

First, add 3 to both sides of the equation to isolate the absolute value expression:
$|x+7| - 3 = 8$ $|x+7| = 8 + 3$ $|x+7| = 11$

STEP 4

The absolute value equation $|x+7| = 11$ implies two separate equations:
Case 1: $x+7 = 11$
Case 2: $x+7 = -11$

STEP 5

Solve Case 1:
$x + 7 = 11$
Subtract 7 from both sides:
$x = 11 - 7$ $x = 4$

STEP 6

Solve Case 2:
$x + 7 = -11$
Subtract 7 from both sides:
$x = -11 - 7$ $x = -18$

STEP 7

Verify the solutions by substituting back into the original equation:
For $x = 4$ :
$|4 + 7| - 3 = |11| - 3 = 11 - 3 = 8$ (True)
For $x = -18$ :
$|-18 + 7| - 3 = |-11| - 3 = 11 - 3 = 8$ (True)
Both solutions satisfy the original equation.
The solutions for $x$ are:
$\boxed{4} \quad \text{and} \quad \boxed{-18}$

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