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Chapter 0: Problem 79

Solve each absolute value inequality. $$-2|x-4| \geq-4$$

Short Answer

Expert verified
The solution to the inequality is \(2 \leq x \leq 6\)

Step by step solution

01

Isolate the absolute value

To isolate the absolute value, divide both sides of the inequality by -2. Note that dividing by a negative number reverses the inequality symbol. Hence, \( |x-4| \leq 2 \).
02

Split into two cases

Now split the inequality into two separate cases according to the definition of the absolute value: Case 1: \(x - 4 \leq 2\) Case 2: -(x - 4) \leq 2
03

Solve each case

For case 1: adds 4 to both sides to get \(x \leq 6\). For case 2: multiply by -1 on both sides (which also reverse the inequality) to get \(x - 4 \geq -2\), then add 4 to both sides to get \(x \geq 2\).
04

Combine the solution sets

By considering the solution from both cases, conclude that the solution to the original inequality is \(2 \leq x \leq 6\)

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Most popular questions from this chapter

Rationalize the numerator. $$\frac{\sqrt{x}+\sqrt{y}}{x^{2}-y^{2}}$$

If you are given a quadratic equation, how do you determine which method to use to solve it?

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The rational expressions $$\frac{7}{14 x} \text { and } \frac{7}{14+x}$$ can both be simplified by dividing each numerator and each denominator by 7

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In an inequality such as \(5 x+4<8 x-5,\) I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.
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