Question

Solve the following inequality. Graph and check your solution.
15 z less than minus 75
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Part 1
Solve the inequality.
  
enter your response here
​(Simplify your answer. Type an​ inequality.)

Answer 1

To solve the inequality \(15z < -75\), we can isolate \(z\) by dividing both sides by 15. Here's how we do it step by step:

1. Start with the inequality:
\[
15z < -75
\]

2. Divide both sides by 15:
\[
z < \frac{-75}{15}
\]

3. Simplify the right side:
\[
z < -5
\]

Thus, the solution to the inequality is:
\[
z < -5
\]

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### Graphing the Solution

To graph the solution \(z < -5\):

1. Draw a number line.
2. Locate the point -5 on the number line.
3. Since the inequality is strict (the "<" symbol does not include -5), use an open circle at -5.
4. Shade the line to the left to indicate all numbers less than -5.

### Checking the Solution

To verify the solution, we can pick a number less than -5 and a number greater than -5:

- **Choose \(z = -6\) (which is less than -5):**
\[
15(-6) = -90 < -75 \quad (\text{True})
\]

- **Choose \(z = -4\) (which is greater than -5):**
\[
15(-4) = -60 < -75 \quad (\text{False})
\]

Since our check holds true for values less than -5 and false for values greater than -5, we can confirm that the solution is correctly represented by \(z < -5\).