### Part A: Solving the Inequality
We are given the inequality:
[tex]\[ \frac{2}{3} - \frac{1}{2} p \leq \frac{8}{3} \][/tex]
Step 1: Isolate the term involving [tex]\( p \)[/tex]
First, we need to move [tex]\( \frac{2}{3} \)[/tex] to the other side by subtracting [tex]\( \frac{2}{3} \)[/tex] from both sides:
[tex]\[ \frac{2}{3} - \frac{1}{2} p - \frac{2}{3} \leq \frac{8}{3} - \frac{2}{3} \][/tex]
Simplifying both sides:
[tex]\[ -\frac{1}{2} p \leq \frac{8}{3} - \frac{2}{3} \][/tex]
Calculate [tex]\( \frac{8}{3} - \frac{2}{3} \)[/tex]:
[tex]\[ \frac{8}{3} - \frac{2}{3} = \frac{8 - 2}{3} = \frac{6}{3} = 2 \][/tex]
So, the inequality simplifies to:
[tex]\[ -\frac{1}{2} p \leq 2 \][/tex]
Step 2: Solve for [tex]\( p \)[/tex]
To isolate [tex]\( p \)[/tex], multiply both sides of the inequality by -2. Remember, multiplying or dividing by a negative number reverses the inequality sign:
[tex]\[ -\frac{1}{2} p \times (-2) \geq 2 \times (-2) \][/tex]
This simplifies to:
[tex]\[ p \geq -4 \][/tex]
So, the solution to the inequality is:
[tex]\[ p \geq -4 \][/tex]
### Part B: Graphing the Solution on a Number Line
To graph the inequality [tex]\( p \geq -4 \)[/tex] on a number line, follow these steps:
1. Draw a horizontal line to represent the number line.
2. Find the point -4 on the number line and make a solid circle at this point to indicate that [tex]\( p \)[/tex] can be equal to -4.
3. Shade the number line to the right of -4 to indicate that [tex]\( p \)[/tex] includes all the numbers greater than -4.
By shading to the right, we show that [tex]\( p \)[/tex] can be any number greater than or equal to -4. This graphical representation gives a clear visual understanding of the range of values that [tex]\( p \)[/tex] can take.
