Solve each equation or inequality. | 10- 4x | ≥ 4

Verified step by step guidance

1. Start by understanding the absolute value inequality: \(|10 - 4x| \geq 4\). This means the expression inside the absolute value, \(10 - 4x\), is either greater than or equal to 4, or less than or equal to -4.
2. Set up the two separate inequalities: \(10 - 4x \geq 4\) and \(10 - 4x \leq -4\).
3. Solve the first inequality: \(10 - 4x \geq 4\). Subtract 10 from both sides to get \(-4x \geq -6\). Then, divide both sides by -4, remembering to flip the inequality sign, resulting in \(x \leq \frac{3}{2}\).
4. Solve the second inequality: \(10 - 4x \leq -4\). Subtract 10 from both sides to get \(-4x \leq -14\). Then, divide both sides by -4, remembering to flip the inequality sign, resulting in \(x \geq \frac{7}{2}\).
5. Combine the solutions from both inequalities. The solution to the original inequality \(|10 - 4x| \geq 4\) is \(x \leq \frac{3}{2}\) or \(x \geq \frac{7}{2}\).

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 equations and inequalities that involve it, as it leads to two separate cases to consider.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal, using symbols such as ≥, ≤, >, or <. When solving inequalities, it is important to maintain the direction of the inequality when performing operations, especially when multiplying or dividing by negative numbers. This concept is essential for determining the solution set of the given absolute value inequality.

Case Analysis

Case analysis involves breaking down a problem into multiple scenarios based on different conditions. In the context of absolute value inequalities, this means creating separate equations for the positive and negative cases of the expression inside the absolute value. This method allows for a comprehensive solution that accounts for all possible values that satisfy the original inequality.