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Chapter 1: Problem 72

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

Short Answer

Expert verified
The solution to the given absolute value inequality \(|x+3| \geq 4\) is \(x \leq -7\) or \(x \geq 1\). In interval notation, this is \((-\infty, -7] \cup [1, \infty)\).

Step by step solution

01

Understand the Absolute Value Property

The absolute value inequality \(|x+3| \geq 4\) means that whatever is inside the absolute value (here \(x + 3\)) is either greater than 4 or less than -4.
02

Split the Inequality

Split the absolute inequality into two inequalities. The first inequality is \(x + 3 \geq 4\) and the second inequality is \(-(x + 3)\geq 4\).
03

Solve the First Inequality

Solve the first inequality \(x + 3 \geq 4\) by subtracting 3 from both sides to come up with \(x \geq 1\).
04

Solve the Second Inequality

Solve the second inequality \(-(x + 3) \geq 4\) by distributing the negative sign to get \(-x - 3 \geq 4\). From this, add 3 to both sides to get \(-x \geq 7\). Lastly, multiply by -1 to both sides, remembering to flip the inequality sign due to the multiplication of the negative number. This will result in \(x \leq -7\).
05

Write the Solution

Combine the two solutions, use the word 'or' to indicate that either solution satisfies the original inequality. The mathematical notation to express this in interval notation is: \((-\infty, -7] \cup [1, \infty)\).

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