Question

solve the inequality and graph the solution.

$1 > \frac{z}{3} - 1$

plot the endpoints. select an endpoint to change it from closed to open. select the middle of the segment, ray, or line to delete it.

<---(-10)---(-8)---(-6)---(-4)---(-2)---(0)---(2)---(4)---(6)---(8)---(10)--->

Explanation:

Step1: Add 1 to both sides

To isolate the term with \( z \), we add 1 to both sides of the inequality \( 1 > \frac{z}{3} - 1 \). This gives \( 1 + 1 > \frac{z}{3} - 1 + 1 \), which simplifies to \( 2 > \frac{z}{3} \).

Step2: Multiply both sides by 3

To solve for \( z \), we multiply both sides of the inequality \( 2 > \frac{z}{3} \) by 3. This gives \( 2\times3 > \frac{z}{3}\times3 \), which simplifies to \( 6 > z \) or \( z < 6 \).

To graph this solution:

• The endpoint is at 6. Since the inequality is strict (\( z < 6 \), not \( z \leq 6 \)), we use an open circle at 6.
• Then we draw a ray pointing to the left from 6 (because \( z \) is less than 6) on the number line.

Answer:

The solution to the inequality is \( z < 6 \). The graph has an open circle at 6 and a ray extending to the left from 6.