To solve the inequality \( 14z < -84 \), you can follow these steps:
- Divide both sides by 14 to isolate \( z \). When you divide or multiply both sides of an inequality by a positive number, the direction of the inequality remains unchanged.
\[ z < \frac{-84}{14} \]
- Simplify the right side:
\[ z < -6 \]
So, the solution to the inequality is:
\[ z < -6 \]
Graphing the Solution
To graph the solution on a number line:
- Draw a horizontal line (the number line).
- Mark the point \( -6 \) on the line.
- Since the inequality is \( z < -6 \), you will use an open circle at \( -6 \) (because \( -6 \) is not included).
- Shade the line to the left of \( -6 \) to indicate all numbers less than \( -6 \).
Summary
The solution to the inequality is:
\[ z < -6 \]
You may check your solution by choosing test points. For instance, pick \( z = -7 \) (which is less than \(-6\)) and see if it satisfies the inequality:
\[ 14(-7) < -84 \implies -98 < -84 \quad \text{(True)} \]
Now, pick \( z = -5 \) (which is greater than \(-6\)):
\[ 14(-5) < -84 \implies -70 < -84 \quad \text{(False)} \]
This confirms that the solution \( z < -6 \) is correct.