Solve the inequality and graph the solution.

Question

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7(k–7)+9≥100
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.

Answer 1

To solve the inequality \(-7(k - 7) + 9 \geq 100\), we will first simplify the inequality step by step.

1. Distribute \(-7\) in \(-7(k - 7)\): \[ -7k + 49 + 9 \geq 100 \]

2. Combine like terms: \[ -7k + 58 \geq 100 \]

3. Subtract 58 from both sides: \[ -7k \geq 100 - 58 \] \[ -7k \geq 42 \]

4. Divide both sides by \(-7\). Remember that dividing by a negative number flips the inequality sign: \[ k \leq -6 \]

The solution to the inequality is: \[ k \leq -6 \]

Graphing the Solution

1. Endpoints: The endpoint is \(k = -6\).
2. Changing the endpoint from closed to open: Since the inequality is \(\leq\), we can keep it closed (include \(-6\)).
3. Select the middle of the segment: The solution set extends to the left of \(-6\), and thus we don't change any section; we take the entire ray from \(-\infty\) to \(-6\).

To graph it:

• Plot the closed circle at \(-6\):
• Shade to the left of \(-6\) indicating all values less than or equal to \(-6\).

The graph looks like this on a number line:

<======● -6

Where:

• ● indicates a closed circle at \(-6\), meaning it's included in the solution.
• The arrow to the left indicates all values less than \(-6\) are included.