Solve `abs(x - 13)gt5` for x.

Options

A. 18
B. 8 or 18
C. `x lt 8 or x gt18`
D. 8

Answer

To solve the inequality \( |x - 13| > 5 \), we will break it down into two separate cases based on the definition of absolute value.

### Step 1: Set up the cases
The expression \( |x - 13| > 5 \) means that the distance between \( x \) and \( 13 \) is greater than \( 5 \). This leads us to two cases:
1. \( x - 13 > 5 \)
2. \( -(x - 13) > 5 \) (which simplifies to \( x - 13 < -5 \))

### Step 2: Solve the first case
For the first case:
\[ x - 13 > 5 \]
Add \( 13 \) to both sides:
\[ x > 5 + 13 \]
\[ x > 18 \]

### Step 3: Solve the second case
For the second case:
\[ -(x - 13) > 5 \]
This can be rewritten as:
\[ -1(x - 13) > 5 \]
Multiplying both sides by \(-1\) (remember to reverse the inequality):
\[ x - 13 < -5 \]
Now, add \( 13 \) to both sides:
\[ x < -5 + 13 \]
\[ x < 8 \]

### Step 4: Combine the results
From the two cases, we have:
1. \( x > 18 \)
2. \( x < 8 \)

Thus, the solution to the inequality \( |x - 13| > 5 \) is:
\[ x < 8 \quad \text{or} \quad x > 18 \]

### Final Answer:
The solution set is \( x < 8 \) or \( x > 18 \).

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