Question

The inequality −7(x − 4) < 84 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:

To solve the inequality −7(x − 4) < 84, we will start by distributing the -7 to the terms inside the parentheses:

-7x + 28 < 84

Next, we will subtract 28 from both sides of the inequality to isolate the variable:

-7x < 84 - 28

-7x < 56

Now, we will divide both sides of the inequality by -7. When dividing by a negative number, we need to reverse the inequality symbol:

x > 56/-7

x > -8

So, the solution to the inequality is x > -8.

Part B:

To graph the solution to the inequality x > -8 on a number line, we will start by drawing a number line with a point labeled -8. This point will be an open circle since x is not equal to -8. Then, we will draw an arrow to the right, indicating that all values greater than -8 are solutions to the inequality.

Part C:

Two values that would make the inequality x > -8 true are x = -7 and x = 0.

For x = -7:
Plugging this value into the original inequality, we get:
-7(-7 - 4) < 84
-7(-11) < 84
77 < 84

Since 77 is indeed less than 84, the inequality is true for x = -7.

For x = 0:
Plugging this value into the original inequality, we get:
-7(0 - 4) < 84
-7(-4) < 84
28 < 84

Since 28 is indeed less than 84, the inequality is true for x = 0.

Therefore, x = -7 and x = 0 are two values that make the inequality −7(x − 4) < 84 true.