To graph the solution to the inequality [tex]\( |w + 5| \geq 2 \)[/tex] on the number line, we need to break it down step-by-step:
### Step 1: Understand the Absolute Value Inequality
The inequality [tex]\( |w + 5| \geq 2 \)[/tex] can be interpreted as saying that the distance between [tex]\( w + 5 \)[/tex] and 0 is at least 2.
### Step 2: Break into Two Sub-Inequalities
An absolute value inequality [tex]\( |A| \geq B \)[/tex] for [tex]\( B \geq 0 \)[/tex] translates into:
[tex]\[ A \leq -B \quad \text{or} \quad A \geq B \][/tex]
Applying this to our inequality:
[tex]\[ w + 5 \leq -2 \quad \text{or} \quad w + 5 \geq 2 \][/tex]
### Step 3: Solve Each Sub-Inequality
1. For [tex]\( w + 5 \leq -2 \)[/tex]:
[tex]\[ w \leq -2 - 5 \][/tex]
[tex]\[ w \leq -7 \][/tex]
2. For [tex]\( w + 5 \geq 2 \)[/tex]:
[tex]\[ w \geq 2 - 5 \][/tex]
[tex]\[ w \geq -3 \][/tex]
### Step 4: Combine the Solutions
The two inequalities [tex]\( w \leq -7 \)[/tex] and [tex]\( w \geq -3 \)[/tex] describe the regions where [tex]\( w \)[/tex] satisfies the original inequality. Thus, the solution set is:
[tex]\[ w \leq -7 \quad \text{or} \quad w \geq -3 \][/tex]
### Step 5: Graph on the Number Line
To graph this on the number line:
1. Draw a number line.
2. Mark the points [tex]\(-7\)[/tex] and [tex]\(-3\)[/tex].
3. Shade the region to the left of [tex]\(-7\)[/tex] to represent [tex]\( w \leq -7 \)[/tex]. Use a closed circle at [tex]\(-7\)[/tex] because the inequality includes [tex]\(-7\)[/tex] (i.e., [tex]\(\leq\)[/tex]).
4. Shade the region to the right of [tex]\(-3\)[/tex] to represent [tex]\( w \geq -3 \)[/tex]. Use a closed circle at [tex]\(-3\)[/tex] because the inequality includes [tex]\(-3\)[/tex] (i.e., [tex]\(\geq\)[/tex]).
Here is the number line representation:
[tex]\[
\begin{array}{cccccccccccccccccccccccc}
\text{...}&\underline{\cdot}&\cdot&\cdot&\cdot&\cdot&\cdot&\underline{\circ}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\underline{\ \cdot}&\cdot&\underline{\bullet}&\rightarrow\\
&-7&&&&&-3\\
\end{array}
\][/tex]
This graph illustrates the values of [tex]\( w \)[/tex] that satisfy the inequality [tex]\( |w + 5| \geq 2 \)[/tex].
