Question

The inequality −8(x − 1) ≥ 88 is given.

Part A: Solve the inequality and show every step of your work. (4 points)

Part B: Explain in words how to graph the solution to the inequality on a number line. (4 points)

Part C: Find two values that would make the inequality true. Explain how you know the values are solutions to the inequality. (4 points)

Answer 1

### 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:**

1. For \(x = -11\):

\[
-8(-11 - 1) \geq 88
\]
\[
-8(-12) \geq 88
\]
\[
96 \geq 88 \quad \text{(True)}
\]

2. 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 \(x \leq -10\).