If x is a real number and `|5-(x-3)|+8lt15`, then (i) `1 le x le 15` (ii) `1lt x lt 15` (iii) `1 lt x le 15` (iv) `1 le x le 15`

Options

A. `l le x le 15`
B. `1lt x lt 15`
C. `1 lt x le 15`
D. `1 le x le 15`

Answer

To solve the inequality \( |5 - (x - 3)| + 8 < 15 \), we will follow these steps:

### Step 1: Simplify the Inequality
Start with the given inequality:
\[ |5 - (x - 3)| + 8 < 15 \]
Subtract 8 from both sides:
\[ |5 - (x - 3)| < 15 - 8 \]
This simplifies to:
\[ |5 - (x - 3)| < 7 \]

### Step 2: Rewrite the Absolute Value
Now, we can rewrite the expression inside the absolute value:
\[ |5 - (x - 3)| = |8 - x| \]
So, the inequality becomes:
\[ |8 - x| < 7 \]

### Step 3: Remove the Absolute Value
The inequality \( |8 - x| < 7 \) implies two cases:
1. \( 8 - x < 7 \)
2. \( 8 - x > -7 \)

### Step 4: Solve the First Case
For the first case:
\[ 8 - x < 7 \]
Subtract 8 from both sides:
\[ -x < -1 \]
Multiplying both sides by -1 (which reverses the inequality):
\[ x > 1 \]

### Step 5: Solve the Second Case
For the second case:
\[ 8 - x > -7 \]
Subtract 8 from both sides:
\[ -x > -15 \]
Multiplying both sides by -1 (which reverses the inequality):
\[ x < 15 \]

### Step 6: Combine the Results
From the two cases, we have:
\[ 1 < x < 15 \]
This can be written in interval notation as:
\[ (1, 15) \]

### Conclusion
Thus, the solution to the inequality is:
\[ 1 < x < 15 \]
This corresponds to option (ii) \( 1 < x < 15 \).