Absolute Value Inequality - Number Line worksheet - Free Printable

Problem: Solve the absolute value inequalities and graph the solutions.

#### Step 1: Solve the first inequality \( |x - 4| \leq 1 \)

##### Case 1: \( x - 4 \leq 1 \)
1. Start with the inequality:
\[
x - 4 \leq 1
\]
2. Add 4 to both sides:
\[
x \leq 5
\]

##### Case 2: \( x - 4 \geq -1 \)
1. Start with the inequality:
\[
x - 4 \geq -1
\]
2. Add 4 to both sides:
\[
x \geq 3
\]

##### Combine the results:
The solution to \( |x - 4| \leq 1 \) is:
\[
3 \leq x \leq 5
\]

##### Graph the solution:
- On the number line, shade the interval from 3 to 5, including both endpoints (closed circles at 3 and 5).

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#### Step 2: Solve the second inequality \( |x - 3| \geq 3 \)

##### Case 1: \( x - 3 \geq 3 \)
1. Start with the inequality:
\[
x - 3 \geq 3
\]
2. Add 3 to both sides:
\[
x \geq 6
\]

##### Case 2: \( x - 3 \leq -3 \)
1. Start with the inequality:
\[
x - 3 \leq -3
\]
2. Add 3 to both sides:
\[
x \leq 0
\]

##### Combine the results:
The solution to \( |x - 3| \geq 3 \) is:
\[
x \leq 0 \quad \text{or} \quad x \geq 6
\]

##### Graph the solution:
- On the number line, shade the intervals \( (-\infty, 0] \) and \( [6, \infty) \), including the endpoints (closed circles at 0 and 6).

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Final Answers:

1. For \( |x - 4| \leq 1 \):
\[
\boxed{3 \leq x \leq 5}
\]

2. For \( |x - 3| \geq 3 \):
\[
\boxed{x \leq 0 \quad \text{or} \quad x \geq 6}
\]

Graphs:

- For \( |x - 4| \leq 1 \): Shade the interval from 3 to 5 on the number line.
- For \( |x - 3| \geq 3 \): Shade the intervals \( (-\infty, 0] \) and \( [6, \infty) \) on the number line.