In Exercises 59–94, solve each absolute value inequality. - 2|x - 4| ≥ - 4

Verified step by step guidance

1. Start by isolating the absolute value expression. The inequality is given as \(-2|x - 4| \geq -4\). To isolate \(|x - 4|\), first divide both sides of the inequality by \(-2\). Remember, dividing by a negative number reverses the inequality sign.
2. After dividing both sides by \(-2\), the inequality becomes \(|x - 4| \leq 2\). This means the absolute value of \(x - 4\) is less than or equal to 2.
3. Recall that the definition of absolute value inequality \(|A| \leq B\) (where \(B \geq 0\)) can be rewritten as a compound inequality: \(-B \leq A \leq B\). Apply this to \(|x - 4| \leq 2\) to get \(-2 \leq x - 4 \leq 2\).
4. Next, solve the compound inequality for \(x\) by adding 4 to all three parts: \(-2 + 4 \leq x - 4 + 4 \leq 2 + 4\), which simplifies to \$2 \leq x \leq 6$.
5. The solution to the inequality is all \(x\) values between 2 and 6, inclusive. This can be expressed in interval notation as \([2, 6]\).

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always yielding a non-negative result. For any expression |A|, it equals A if A ≥ 0, and -A if A < 0. Understanding this helps in rewriting and solving absolute value inequalities.

Properties of Inequalities

Inequalities describe the relative size or order of two values. When solving inequalities involving absolute values, it's important to consider the direction of the inequality and how multiplying or dividing by negative numbers reverses the inequality sign.

Solving Absolute Value Inequalities

To solve inequalities like |x - a| ≥ b, where b ≥ 0, split the inequality into two cases: x - a ≥ b or x - a ≤ -b. This approach transforms the absolute value inequality into two linear inequalities that can be solved separately.