Sure, let's analyze the inequality [tex]\(|u - 3| > 5\)[/tex] step-by-step and then graph the solution on the number line.
### Step-by-Step Solution
1. Understanding Absolute Value Inequality:
The inequality [tex]\(|u - 3| > 5\)[/tex] translates to two separate inequalities:
[tex]\[
u - 3 > 5 \quad \text{or} \quad u - 3 < -5
\][/tex]
2. Solve Each Inequality Separately:
- For the inequality [tex]\(u - 3 > 5\)[/tex]:
[tex]\[
u - 3 > 5
\][/tex]
Adding 3 to both sides:
[tex]\[
u > 8
\][/tex]
- For the inequality [tex]\(u - 3 < -5\)[/tex]:
[tex]\[
u - 3 < -5
\][/tex]
Adding 3 to both sides:
[tex]\[
u < -2
\][/tex]
3. Combine the Solution Intervals:
- From [tex]\(u - 3 > 5\)[/tex]: [tex]\(u > 8\)[/tex]
- From [tex]\(u - 3 < -5\)[/tex]: [tex]\(u < -2\)[/tex]
Therefore, the combined solution is:
[tex]\[
u < -2 \quad \text{or} \quad u > 8
\][/tex]
### Graphing the Solution
To represent this on a number line:
1. Draw a horizontal line, which will be our number line.
2. Mark the critical points [tex]\(u = -2\)[/tex] and [tex]\(u = 8\)[/tex].
3. Shade the regions that satisfy the solution:
- Shade the region to the left of [tex]\(-2\)[/tex] (not including [tex]\(-2\)[/tex]).
- Shade the region to the right of [tex]\(8\)[/tex] (not including [tex]\(8\)[/tex]).
4. Use open circles at [tex]\(u = -2\)[/tex] and [tex]\(u = 8\)[/tex] to indicate that these points are not included in the solution.
Here’s how it looks:
[tex]\[
\begin{array}{lcr}
\text{\(-\infty\)} & \text{-2} & \text{8} & \text{\(\infty\)} \\
\text{(} & \text{)} & \text{(} & \text{)} \\
\text{\(-\infty\) < \(u\)} & \text{\(< -2\)} & \text{\(u >\)} & \text{\(8 < \infty\)}
\end{array}
\][/tex]
The graph can be visualized as follows:
```
---o=========================o---
-2 8
```
- The region to the left of [tex]\(-2\)[/tex] (shaded, but not including [tex]\(-2\)[/tex])
- The region to the right of [tex]\(8\)[/tex] (shaded, but not including [tex]\(8\)[/tex])
This is the graphical representation of the solution to the inequality [tex]\(|u - 3| > 5\)[/tex].
