Question

Solve the absolute value equation or indicate that the equation has no solution.
StartAbsoluteValue x minus 3 EndAbsoluteValue equals 5

Answer 1

To solve the absolute value equation \( |x - 3| = 5 \), we need to consider the two cases that arise from the definition of absolute value.

1. **Case 1**: The expression inside the absolute value is positive or zero:
\[
x - 3 = 5
\]
To solve for \( x \), add 3 to both sides:
\[
x = 5 + 3 = 8
\]

2. **Case 2**: The expression inside the absolute value is negative:
\[
x - 3 = -5
\]
To solve for \( x \), again add 3 to both sides:
\[
x = -5 + 3 = -2
\]

Therefore, the solutions to the equation \( |x - 3| = 5 \) are:
\[
x = 8 \quad \text{and} \quad x = -2
\]

In conclusion, the solutions are \( x = 8 \) and \( x = -2 \).