Solve each equation or inequality. | 10 - 4x | + 1 ≥ 5

Verified step by step guidance

1. Start by isolating the absolute value expression: subtract 1 from both sides to get |10 - 4x| ≥ 4.
2. Recognize that the inequality |A| ≥ B means A ≤ -B or A ≥ B.
3. Apply this to the expression: 10 - 4x ≤ -4 or 10 - 4x ≥ 4.
4. Solve the first inequality: 10 - 4x ≤ -4 by subtracting 10 from both sides and then dividing by -4, remembering to reverse the inequality sign.
5. Solve the second inequality: 10 - 4x ≥ 4 by subtracting 10 from both sides and then dividing by -4.

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 can lead 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 sign, especially when multiplying or dividing by negative numbers. This concept is essential for determining the solution set of the given inequality.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. This process often requires isolating the variable on one side of the equation through various algebraic manipulations. In the context of the given problem, solving the absolute value inequality will involve breaking it down into two separate cases based on the definition of absolute value.