Question

question

1. solve the equation $\frac{1}{2}x - 1=| x - 5 |$

answer attempt 1 out of 2

Explanation:

Step1: Define absolute value cases

For \(|x-5|\), two cases:
Case 1: \(x - 5 \geq 0 \implies x \geq 5\), so \(|x-5|=x-5\)
Case 2: \(x - 5 < 0 \implies x < 5\), so \(|x-5|=-(x-5)=5-x\)

Step2: Solve Case 1 (\(x \geq 5\))

Set \(\frac{1}{2}x - 1 = x - 5\)
Rearrange terms:
\(-1 + 5 = x - \frac{1}{2}x\)
\(4 = \frac{1}{2}x\)
Multiply both sides by 2: \(x = 8\)
Verify: \(8 \geq 5\), valid.

Step3: Solve Case 2 (\(x < 5\))

Set \(\frac{1}{2}x - 1 = 5 - x\)
Rearrange terms:
\(\frac{1}{2}x + x = 5 + 1\)
\(\frac{3}{2}x = 6\)
Multiply both sides by \(\frac{2}{3}\): \(x = 6 \times \frac{2}{3} = 4\)
Verify: \(4 < 5\), valid.

Answer:

\(x = 4\) and \(x = 8\)