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

Verified step by step guidance

1. Start by understanding the absolute value inequality: \(5 > |4 - x|\). This means the expression inside the absolute value, \(4 - x\), must be between \(-5\) and \(5\).
2. Set up two separate inequalities to remove the absolute value: \(-5 < 4 - x < 5\).
3. Solve the left inequality: \(-5 < 4 - x\). Add \(x\) to both sides to get \(x - 5 < 4\). Then add \(5\) to both sides to isolate \(x\), resulting in \(x < 9\).
4. Solve the right inequality: \(4 - x < 5\). Subtract \(4\) from both sides to get \(-x < 1\). Multiply both sides by \(-1\) (and remember to flip the inequality sign) to get \(x > -1\).
5. Combine the results from both inequalities to find the solution set: \(-1 < x < 9\). This is the range of values for \(x\) that satisfy the original inequality.

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.