Math

Question Solve the absolute value equation x8+4=5\left|\frac{x}{8}\right|+4=5 for the value of xx.

Studdy Solution

STEP 1

Assumptions
1. We are solving the equation x8+4=5\left|\frac{x}{8}\right|+4=5 for the variable xx.
2. The absolute value function y|y| is defined such that y=y|y| = y if y0y \geq 0 and y=y|y| = -y if y<0y < 0.

STEP 2

First, we need to isolate the absolute value expression on one side of the equation. To do this, we subtract 4 from both sides of the equation.
x8+44=54\left|\frac{x}{8}\right|+4-4=5-4

STEP 3

Simplify both sides of the equation.
x8=1\left|\frac{x}{8}\right|=1

STEP 4

Now, we need to consider the two cases for the absolute value function. The first case is when the expression inside the absolute value is non-negative, and the second case is when it is negative.
Case 1: x80\frac{x}{8} \geq 0
Case 2: x8<0\frac{x}{8} < 0

STEP 5

Solve for xx in Case 1, where x8\frac{x}{8} is non-negative.
x8=1\frac{x}{8} = 1

STEP 6

Multiply both sides of the equation by 8 to solve for xx.
x=1×8x = 1 \times 8

STEP 7

Calculate the value of xx for Case 1.
x=8x = 8

STEP 8

Solve for xx in Case 2, where x8\frac{x}{8} is negative.
x8=1-\frac{x}{8} = 1

STEP 9

Multiply both sides of the equation by -8 to solve for xx.
x=1×8x = -1 \times 8

STEP 10

Calculate the value of xx for Case 2.
x=8x = -8
The solution to the equation x8+4=5\left|\frac{x}{8}\right|+4=5 is x=8x = 8 or x=8x = -8.

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