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

Part A: Solve the inequality and show every step of your work.

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

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

1 answer

Part A:

To solve the inequality −8(x − 1) ≥ 88, let's simplify it step-by-step using the properties of inequalities.

Step 1: Distribute -8 to (x - 1)

-8(x - 1) ≥ 88
-8x + 8 ≥ 88

Step 2: Move the constant term to the right side by subtracting 8 from both sides of the inequality.

-8x + 8 - 8 ≥ 88 - 8
-8x ≥ 80

Step 3: Divide both sides by -8, remembering to reverse the inequality sign when dividing by a negative number.

(-8x)/(-8) ≤ 80/(-8)
x ≤ -10

So the solution to the inequality is x ≤ -10.

Part B:

To graph the solution on a number line, we start by drawing a number line and marking -10 on it. Since the solution is x ≤ -10, we shade everything to the left of -10 to show that all values less than or equal to -10 are solutions. We can also put an arrow pointing to the left starting from -10 to represent the infinite values less than -10.

Number line:
<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Solution on the number line: (-∞, -10]

Part C:

To find two values that would make the inequality true, we can choose any value less than or equal to -10.

Let's choose x = -11 and x = -12.

Substituting x = -11 into the original inequality:

−8(x − 1) ≥ 88
−8(-11 − 1) ≥ 88
-8(-12) ≥ 88
96 ≥ 88

Since 96 is indeed greater than or equal to 88, x = -11 is a solution to the inequality.

Substituting x = -12 into the original inequality:

−8(x − 1) ≥ 88
−8(-12 − 1) ≥ 88
-8(-13) ≥ 88
104 ≥ 88

Again, 104 is greater than or equal to 88, so x = -12 is also a solution to the inequality.

We know these values are solutions to the inequality because when we substitute them into the inequality, the resulting inequality is true.
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