Problem 72 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: Split the Inequality

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

Step 3: Solve the First Inequality

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

Step 4: 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 \(x \leq -7\).

Step 5: 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: \((-\infty, -7] \cup [1, \infty)\).