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Chapter 0: Problem 70

Solve each absolute value inequality. $$|x+3| \geq 4$$

Short Answer

Expert verified
The solutions to the absolute value inequality are \( x \leq -7 \) and \( x \geq 1 \).

Step by step solution

01

Establish the two cases

Since we are dealing with an absolute value, we know that the value inside can be positive or negative and still satisfy the inequality. So, for the expression \( |x+3| \geq 4 \), there are two possible 'cases' or scenarios:1. \( x + 3 \geq 4 \)2. \( -(x + 3) \geq 4 \) (this covers the negative scenario)
02

Solve each case separately

Case 1: \( x + 3 \geq 4 \) can be simplified by subtracting 3 from both sides to give: \( x \geq 4 - 3 \)Which simplifies to: \( x \geq 1 \)Case 2: \( -(x + 3) \geq 4 \) can be simplified by distributing the negative sign and subtracting 3 from both sides to give: \( -x - 3 \geq 4 \) After moving \( x \) to the other side and subtracting 4 from both sides, we get: \( x \leq -7 \)
03

Combine the Solutions

According to the inequalities we established, \( x \) is either less than or equal to -7 or greater than equal to 1. Both cases are part of our solution, so the answer is \( x \leq -7, x \geq 1 \). This is the set of base real numbers that make the original inequality true.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When performing the division $$\frac{7 x}{x+3}+\frac{(x+3)^{2}}{x-5}$$ I began by dividing the numerator and the denominator by the common factor, \(x+3\).

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{1}{a^{3}-b^{3}} \cdot \frac{a c+a d-b c-b d}{1}\right)-\frac{c-d}{a^{2}+a b+b^{2}}$$

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

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\).
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