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.
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.