Question
Solve the inequality and graph the solution set. |x-6|<9
Answers
Answered by
GPT-4o mini
To solve the inequality \(|x - 6| < 9\), we need to split it into two linear inequalities based on the definition of absolute value. The expression \(|A| < B\) translates to \( -B < A < B\).
Here, \(A\) is \(x - 6\) and \(B\) is \(9\). Hence, we have:
\[
-9 < x - 6 < 9
\]
Now, we can solve this compound inequality step-by-step.
1. **Add 6 to all parts of the inequality**:
\[
-9 + 6 < x < 9 + 6
\]
This simplifies to:
\[
-3 < x < 15
\]
2. **We can rewrite this in interval notation**:
The solution set is \((-3, 15)\).
**Graphing the solution set:**
On a number line, you would represent this interval as an open interval from \(-3\) to \(15\). This would be shown as:
```
<---|----|----|----|----|----|----|----|----|----|----|----|----|----|--->
-4 -3 -2 -1 0 1 2 3 4 ... 14 15
(-----------------)
```
The parentheses indicate that \(-3\) and \(15\) are not included in the solution set. Thus, the values of \(x\) that satisfy the inequality \(|x - 6| < 9\) are all real numbers between \(-3\) and \(15\), excluding those endpoints.
Here, \(A\) is \(x - 6\) and \(B\) is \(9\). Hence, we have:
\[
-9 < x - 6 < 9
\]
Now, we can solve this compound inequality step-by-step.
1. **Add 6 to all parts of the inequality**:
\[
-9 + 6 < x < 9 + 6
\]
This simplifies to:
\[
-3 < x < 15
\]
2. **We can rewrite this in interval notation**:
The solution set is \((-3, 15)\).
**Graphing the solution set:**
On a number line, you would represent this interval as an open interval from \(-3\) to \(15\). This would be shown as:
```
<---|----|----|----|----|----|----|----|----|----|----|----|----|----|--->
-4 -3 -2 -1 0 1 2 3 4 ... 14 15
(-----------------)
```
The parentheses indicate that \(-3\) and \(15\) are not included in the solution set. Thus, the values of \(x\) that satisfy the inequality \(|x - 6| < 9\) are all real numbers between \(-3\) and \(15\), excluding those endpoints.
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