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Chapter 0: Problem 63

Solve each absolute value inequality. $$|2(x-1)+4| \leq 8$$

Short Answer

Expert verified
The solution set to the absolute value inequality |2(x-1)+4| 鈮� 8 is 1 鈮� x 鈮� 3.

Step by step solution

01

Express the inequality without absolute value

First, we need to express the inequality without the absolute value. We can do this by understanding that the expression inside the absolute value (2(x-1)+4) can either be positive or negative. This gives us two scenarios: \n\nScenario 1: 2(x-1)+4 鈮� 8 \nScenario 2: -(2(x-1)+4) 鈮� 8Both scenarios will yield different x-values, which we would need to combine for the final answer.
02

Solve the first scenario

Now we can solve the first scenario: 2(x-1)+4 鈮� 8. \n\nFirst, simplify the left side: \n2x - 2 + 4 鈮� 8 \n2x + 2 鈮� 8 \n\nThen, isolate x by first subtracting 2 and then dividing by 2: \n2x 鈮� 8 - 2 \n2x 鈮� 6 \nx 鈮� 3
03

Solve the second scenario

Next up is to do the same with the second scenario: -(2(x-1)+4) 鈮� 8. \n\nFirst, simplify the left side: \n-(2x - 2 + 4) 鈮� 8 \n-2x + 6 鈮� 8 \n\nThen, isolate x like before but remember that since we are dealing with a negative x, when we divide by -2, the inequality sign will flip: \n2x 鈮� 8 - 6 \n2x 鈮� 2 \nx 鈮� 1
04

Combine the solutions

The solution to our inequality is the combination of the solutions of both scenarios. This leads us to: \n1 鈮� x 鈮� 3

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A city commission has proposed two tax bills. The first bill requires that a homeowner pay \(\$ 1800\) plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of \(\$ 200\) plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{2 x-1}{x-7}+\frac{3 x-1}{x-7}-\frac{5 x-2}{x-7}=0$$

What is the discriminant and what information does it provide about a quadratic equation?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$x^{2}+36=(x+6)^{2}$$

Describe two ways to simplify \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\).
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