Part A: Solve the inequality and show every step of your work.
We start with the inequality:
\[ -8(x - 1) \geq 88 \]
Step 1: Distribute \(-8\) on the left side.
\[ -8x + 8 \geq 88 \]
Step 2: Subtract \(8\) from both sides to isolate the term with \(x\).
\[ -8x \geq 88 - 8 \] \[ -8x \geq 80 \]
Step 3: Divide both sides by \(-8\). Remember that when you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign.
\[ x \leq \frac{80}{-8} \] \[ x \leq -10 \]
So the solution to the inequality is:
\[ x \leq -10 \]
Part B: Explain in words how to graph the solution to the inequality on a number line.
To graph the solution \(x \leq -10\) on a number line, draw a solid circle (or a filled dot) on the point \(-10\) to indicate that \(-10\) is included in the solution (as the inequality is inclusive). Then shade the line to the left of \(-10\) to show that all numbers less than \(-10\) are also included in the solution.
Part C: Find two values that would make the inequality true. Explain how you know the values are solutions to the inequality.
Two values that would make the inequality true are \(-11\) and \(-20\).
Checking the values:
- For \(x = -11\):
\[ -8(-11 - 1) \geq 88 \] \[ -8(-12) \geq 88 \] \[ 96 \geq 88 \quad \text{(True)} \]
- For \(x = -20\):
\[ -8(-20 - 1) \geq 88 \] \[ -8(-21) \geq 88 \] \[ 168 \geq 88 \quad \text{(True)} \]
Both values satisfy the original inequality, confirming they are solutions because they satisfy \(x \leq -10\).