Let's solve the inequality step-by-step and then graph the solution.
### Step-by-Step Solution
1. Start with the given inequality:
[tex]\[
5x + 17 \geq 52
\][/tex]
2. Isolate the term with the variable by subtracting 17 from both sides:
[tex]\[
5x + 17 - 17 \geq 52 - 17
\][/tex]
3. Simplify the terms:
[tex]\[
5x \geq 35
\][/tex]
4. Divide both sides by 5 to solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{5x}{5} \geq \frac{35}{5}
\][/tex]
5. Simplify the fraction:
[tex]\[
x \geq 7
\][/tex]
### Graphing the Solution on a Number Line
1. Draw a horizontal number line.
2. Identify and mark the point [tex]\( x = 7 \)[/tex] on the number line.
3. Since [tex]\( x \)[/tex] can be any number greater than or equal to 7, use a closed circle at 7 to indicate that 7 is included in the solution set.
4. Shade the region to the right of 7 to represent all numbers greater than 7.
### Final Representation
- Inequality Notation:
[tex]\[
x \geq 7
\][/tex]
- Graph on a Number Line:
```
<---|---|---|---|---|---|---|---|---|---|---|---|---|-->
4 5 6 (7) 8 9 10 11 12 13 14 15
↑
Closed Circle
Shade Right
```
The closed circle at 7 indicates that 7 is included in the solution, and the shaded area to the right of the closed circle represents all numbers greater than or equal to 7.
