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Chapter 1: Problem 81

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

Division

Divide both sides of the inequality by -2. Remember that dividing or multiplying an inequality by a negative number switches the direction of the inequality sign. This gives \(|x-4| \leq 2\).
02

Split the Absolute Value Inequality

Split the absolute value inequality into two separate inequalities: \(x-4 \leq 2\) and \(-(x-4) \leq 2\).
03

Solve each Inequality

Solving the two inequalities will give: For \(x-4 \leq 2\), adding 4 to both sides gives \(x \leq 6\).For \(-(x-4) \leq 2\), distribute the negative, giving \(-x + 4 \leq 2\). Subtracting 4 from both sides gives \(-x \leq -2\). Multiplying each side by -1 (and flipping the inequality sign as per Step 1) gives \(x \geq 2\).
04

Write down the solution

The solutions to the original inequality are values of x that satisfy \(both\) inequalities. That means \(x\) must be between 2 and 6, including the endpoints.

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Most popular questions from this chapter

Did you have difficulties solving some of the problems that were assigned in this Exercise Set? Discuss what you did if this happened to you. Did your course of action enhance your ability to solve algebraic word problems?

Write an original word problem that can be solved using a linear equation. Then solve the problem.

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

Write a quadratic equation in general form whose solution set is \(\\{-3,5\\}\).

In Exercises \(123-124,\) list all numbers that must be excluded from the domain of each rational expression. $$ \frac{3}{2 x^{2}+4 x-9} $$
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