To solve the inequality $|x - 5| \geq 2$ and graph it on the number line, follow these steps:

1. Break the absolute value inequality into two separate inequalities:

• For absolute value inequalities, we handle $|x - 5| \geq 2$ as two cases: $x - 5 \geq 2 \quad \text{or} \quad x - 5 \leq -2$

2. Solve each inequality:

• First case: $x - 5 \geq 2$ $x \geq 7$

• Second case: $x - 5 \leq -2$ $x \leq 3$

3. Combine the solution:

The solution to the inequality is: $x \geq 7 \quad \text{or} \quad x \leq 3$

This represents two disjoint intervals:

• $x \in (-\infty, 3]$
• $x \in [7, \infty)$

4. Graph the solution on the number line:

• At $x = 3$, put a closed dot since $x \leq 3$ includes $3$, and shade the number line to the left towards $-\infty$.
• At $x = 7$, put another closed dot since $x \geq 7$ includes $7$, and shade the number line to the right towards $\infty$.

This graph shows two distinct intervals.

Would you like more details about any of the steps, or do you have any questions?

Related Questions:

1. How do you solve absolute value inequalities when the sign is $<$ instead of $\geq$?
2. What is the general rule for solving $|x - a| \geq b$?
3. How can inequalities be represented using interval notation?
4. What is the difference between open and closed dots on a number line?
5. How does the solution of an absolute value inequality change if we have $>$ instead of $\geq$?

Tip:

When graphing inequalities, remember that a closed dot is used when the variable is included (i.e., $\geq$ or $\leq$), while an open dot is used for strict inequalities (i.e., $>$ or $<$).