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)
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.