Problem 87 - College Algebra Essentials

Question

Solve each absolute value inequality. $$5>|4-x|$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x > -1\) or in interval notation, \((-1, +∞)\).

Step 1: Splitting the inequality

First, we split the inequality into two separate cases: |4-x| < 5 and |4-x| > -5. However, because absolute value is always positive, we can ignore the second case.

Step 2: Solve the inequality

Next, we solve the inequality \(4-x < 5\). We begin by subtracting 4 from both sides of the equation, which gives us: \(-x < 1\). Then, we multiply each side by -1 to solve for x. This results in the inequality \(x > -1\). Remember, when multiplying or dividing by a negative number, the direction of the inequality sign changes.

Step 3: Solution and Interval Notation

Thus, our solution is \(x > -1\). This indicates that all numbers greater than -1 will satisfy our original inequality since all these values are less than 5 units away from 4. In interval notation, this would be expressed as \((-1, +∞)\).