Solve each equation or inequality. | 4 - 4x | + 2 = 4

Verified step by step guidance

1. Start by isolating the absolute value expression. Subtract 2 from both sides of the equation: \(|4 - 4x| + 2 - 2 = 4 - 2\).
2. This simplifies to \(|4 - 4x| = 2\).
3. To solve the absolute value equation, consider the two cases: \(4 - 4x = 2\) and \(4 - 4x = -2\).
4. For the first case, solve \(4 - 4x = 2\) by subtracting 4 from both sides and then dividing by -4.
5. For the second case, solve \(4 - 4x = -2\) by subtracting 4 from both sides and then dividing by -4.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is defined as |x| = x if x ≥ 0 and |x| = -x if x < 0. Understanding absolute value is crucial for solving equations and inequalities that involve expressions within these bars.

Linear Equations

A linear equation is an equation of the first degree, meaning it involves variables raised only to the power of one. The general form is ax + b = c, where a, b, and c are constants. Solving linear equations often involves isolating the variable on one side of the equation, which is essential for finding solutions in the context of the given problem.

Inequalities

Inequalities express a relationship where one side is not necessarily equal to the other, using symbols like <, >, ≤, or ≥. When solving inequalities, the solution set may include a range of values rather than a single number. Understanding how to manipulate and interpret inequalities is important for determining valid solutions in problems involving absolute values.