In Exercises 59–94, solve each absolute value inequality. |(2x + 2)/4| ≥ 2

Verified step by step guidance

1. Start by understanding that the inequality involves an absolute value expression: \(\left| \frac{2x + 2}{4} \right| \geq 2\). The absolute value inequality \(|A| \geq B\) means that either \(A \geq B\) or \(A \leq -B\).
2. Set up two separate inequalities based on the definition of absolute value: 1) \(\frac{2x + 2}{4} \geq 2\) 2) \(\frac{2x + 2}{4} \leq -2\)
3. Solve the first inequality: Multiply both sides by 4 to eliminate the denominator, giving \(2x + 2 \geq 8\). Then isolate \(x\) by subtracting 2 from both sides and dividing by 2.
4. Solve the second inequality similarly: Multiply both sides by 4 to get \(2x + 2 \leq -8\). Then isolate \(x\) by subtracting 2 from both sides and dividing by 2.
5. Combine the solutions from both inequalities to express the solution set for \(x\). Remember, the solution is all \(x\) values that satisfy either inequality.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable or expression is compared to a number. To solve them, consider the definition of absolute value as distance from zero, leading to two cases: one where the expression inside is greater than or equal to the positive value, and one where it is less than or equal to the negative value.

Properties of Inequalities

When solving inequalities, it is important to remember that multiplying or dividing both sides by a negative number reverses the inequality sign. Also, inequalities can be split into compound inequalities when dealing with absolute values, requiring careful handling of each case to find the solution set.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side using algebraic operations such as addition, subtraction, multiplication, or division. The solution is often expressed as an interval or union of intervals, representing all values that satisfy the inequality.