To graph the solution to the inequality [tex]\(|v-3| \geq 6\)[/tex] on the number line, we need to solve the inequality and identify the critical points. Here's a step-by-step solution:
1. Understand the Inequality:
The inequality [tex]\(|v-3| \geq 6\)[/tex] means that the distance between [tex]\(v\)[/tex] and 3 is at least 6 units.
2. Split the Absolute Value Inequality:
We split the absolute value inequality into two separate inequalities:
[tex]\[
v - 3 \geq 6 \quad \text{or} \quad v - 3 \leq -6
\][/tex]
3. Solve Each Inequality Separately:
- First Inequality: [tex]\(v - 3 \geq 6\)[/tex]
[tex]\[
v - 3 \geq 6
\][/tex]
Add 3 to both sides:
[tex]\[
v \geq 9
\][/tex]
- Second Inequality: [tex]\(v - 3 \leq -6\)[/tex]
[tex]\[
v - 3 \leq -6
\][/tex]
Add 3 to both sides:
[tex]\[
v \leq -3
\][/tex]
4. Combine the Solutions:
The solution to the inequality [tex]\( |v-3| \geq 6 \)[/tex] is therefore [tex]\( v \geq 9 \)[/tex] or [tex]\( v \leq -3 \)[/tex].
5. Graph the Solutions on the Number Line:
- For [tex]\( v \geq 9 \)[/tex], we draw a ray starting at 9 and extending to the right (towards positive infinity).
- For [tex]\( v \leq -3 \)[/tex], we draw a ray starting at -3 and extending to the left (towards negative infinity).
Here is the graphical representation on the number line:
[tex]\[
\begin{array}{c|cccccccccccccccccccccccc}
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\end{array}
\][/tex]
[tex]\[
\begin{aligned}
\leftarrow & \text{\,--\,,\,-\,,\,-\,,\,-\,,\,}(-3)\text{\,,\,-\,,\,---\,,\,} \dots &(-2)\text{,,--,}(-1)\text{,,--,,--}\rightarrow & \text{\--,} (8)\text{,,\,,\,}(9) \text{,,\textup{-----},\,,} \dots
\end{aligned}
\][/tex]
Remember that we use solid dots ( ● ) or open dots ( ○ ) on the number line depending on the equality condition. Since the inequality sign in this case is “greater than or equal to” ( ≥ ), we use solid dots at the points -3 and 9 to indicate that these points are included in the solution set. The rays extending from these points indicate all values of [tex]\( v \)[/tex] that satisfy the inequality.
