In Exercises 59–94, solve each absolute value inequality. 5 > |4 - x|

Key Concepts

Absolute Value

Absolute value represents 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 bars.

Inequalities

Inequalities express a relationship between two values that are not necessarily equal, using symbols such as <, >, ≤, or ≥. In the context of absolute value inequalities, we often need to split the inequality into two separate cases to find the range of values that satisfy the condition. This requires careful manipulation and understanding of the properties of inequalities.

Solving Absolute Value Inequalities

To solve an absolute value inequality like 5 > |4 - x|, we first rewrite it as two separate inequalities: 4 - x < 5 and 4 - x > -5. This process involves isolating the variable and determining the solution set for each case. The final solution is typically expressed as an interval or union of intervals that represent all possible values of x that satisfy the original inequality.