If `5lt|2-x|lt6 and xgt0`, what is one possible value of x?

Answer

To solve the inequality \( 5 < |2 - x| < 6 \) with the condition \( x > 0 \), we will break it down step by step.

### Step 1: Break down the absolute value inequality
The inequality \( 5 < |2 - x| < 6 \) can be split into two separate inequalities:
1. \( |2 - x| > 5 \)
2. \( |2 - x| < 6 \)

### Step 2: Solve the first inequality \( |2 - x| > 5 \)
The absolute value inequality \( |2 - x| > 5 \) means that:
- \( 2 - x > 5 \) or
- \( 2 - x < -5 \)

#### Solving \( 2 - x > 5 \):
\[ 2 - x > 5 \implies -x > 3 \implies x < -3 \]

#### Solving \( 2 - x < -5 \):
\[ 2 - x < -5 \implies -x < -7 \implies x > 7 \]

### Step 3: Solve the second inequality \( |2 - x| < 6 \)
The absolute value inequality \( |2 - x| < 6 \) means that:
- \( 2 - x < 6 \) and
- \( 2 - x > -6 \)

#### Solving \( 2 - x < 6 \):
\[ 2 - x < 6 \implies -x < 4 \implies x > -4 \]

#### Solving \( 2 - x > -6 \):
\[ 2 - x > -6 \implies -x > -8 \implies x < 8 \]

### Step 4: Combine the results
From the first inequality \( |2 - x| > 5 \), we have:
1. \( x < -3 \)
2. \( x > 7 \)

From the second inequality \( |2 - x| < 6 \), we have:
1. \( x > -4 \)
2. \( x < 8 \)

### Step 5: Analyze the intervals
Now we combine the results:
- From \( x < -3 \) and \( x > -4 \), the valid interval is \( -4 < x < -3 \) (not relevant since \( x > 0 \)).
- From \( x > 7 \) and \( x < 8 \), the valid interval is \( 7 < x < 8 \).

### Step 6: Find a possible value of \( x \)
Since we need \( x > 0 \), we can choose any value in the interval \( (7, 8) \). A simple choice is:
\[ x = 7.5 \]

### Final Answer
One possible value of \( x \) is \( 7.5 \).

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