To determine the solution to the inequality [tex]\(|x - 13| < 18\)[/tex], let’s break it down step by step.
### Step 1: Understanding the Absolute Value Inequality
The expression [tex]\(|x - 13| < 18\)[/tex] implies that the distance between [tex]\(x\)[/tex] and 13 is less than 18. This can be interpreted as:
[tex]\[ -18 < x - 13 < 18 \][/tex]
### Step 2: Solving the Compound Inequality
We need to solve the inequality in two parts:
#### Part A:
[tex]\[ x - 13 < 18 \][/tex]
To isolate [tex]\(x\)[/tex], we add 13 to both sides:
[tex]\[ x - 13 + 13 < 18 + 13 \][/tex]
[tex]\[ x < 31 \][/tex]
So, one part of the solution is [tex]\(x < 31\)[/tex].
#### Part B:
[tex]\[ -18 < x - 13 \][/tex]
To isolate [tex]\(x\)[/tex], we add 13 to both sides:
[tex]\[ -18 + 13 < x - 13 + 13 \][/tex]
[tex]\[ -5 < x \][/tex]
So, the other part of the solution is [tex]\(-5 < x\)[/tex].
### Step 3: Combining the Results
Combining the results from Part A and Part B, we get:
[tex]\[ -5 < x < 31 \][/tex]
### Step 4: Interpreting the Solution
This inequality means that [tex]\(x\)[/tex] is greater than -5 and less than 31. In other words, [tex]\(x\)[/tex] must satisfy both conditions simultaneously.
### Conclusion
The correct solution is given by the choice that encompasses both conditions together:
[tex]\[ x < 31 \text{ and } x > -5 \][/tex]
Therefore, the correct answer is:
D. [tex]\( x < 31 \)[/tex] and [tex]\( x > -5 \)[/tex]
