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

Verified step by step guidance

1. Start by isolating the absolute value expression: \( \left| \frac{2x + 2}{4} \right| \geq 2 \).
2. Rewrite the inequality as two separate inequalities: \( \frac{2x + 2}{4} \geq 2 \) or \( \frac{2x + 2}{4} \leq -2 \).
3. Solve the first inequality: Multiply both sides by 4 to eliminate the fraction, resulting in \( 2x + 2 \geq 8 \).
4. Solve the second inequality: Multiply both sides by 4 to eliminate the fraction, resulting in \( 2x + 2 \leq -8 \).
5. Solve each inequality for \( x \) by isolating \( x \) in each case, and then combine the solutions to find the solution set for the original inequality.

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x| and is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. Understanding absolute value is crucial for solving inequalities that involve expressions within absolute value symbols.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be strict (using < or >) or non-strict (using ≤ or ≥). When solving absolute value inequalities, it is important to consider both the positive and negative scenarios that arise from the definition of absolute value, leading to two separate inequalities to solve.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. In the context of absolute value inequalities, this often requires isolating the variable and simplifying the resulting expressions. Mastery of linear equations is essential, as the solutions to the inequalities will typically involve linear expressions that need to be solved for the variable.