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Solve the following inequality. Graph and check your solution.

15 z less than minus 75
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Part 1
Solve the inequality.
  
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​(Simplify your answer. Type an​ inequality.)

1 answer

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: \[ z < -5 \]

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 \(z < -5\).