Problem 117 - College Algebra

Question

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.

Step-by-Step Solution

Answer

The steps to solve an absolute value inequality involve breaking it into two separate equations, one positive and one negative, and then solving those equations to find the solutions. For the equation |x+3| > 7, the solutions are x > 4 and x < -10.

Step 1: Understand the Absolution Value and Nature of Inequality

The absolute value is the distance of the number from zero, regardless of the direction. So, the absolute value of -3 would be 3, and absolute value of 3 would also be 3. The inequality symbol \(>\) means 'greater than'.

Step 2: Set Up the Inequality

Let's take an example of the absolute value inequality |x+3| > 7. Firstly, we must remember that whatever is smaller than -7, or larger than 7, is the solution to this inequality. This is because their distance from zero is more than seven.

Step 3: Breakdown the Inequality

Next, we split the equation into two. Once into a regular equation x + 3 > 7(regular solution), and another one into a negative equation -(x + 3) > 7, because the absolute value indicates the number could have been originally negative.

Step 4: Solve Both Equations

After solving both equations, we find the solutions as x > 4 for the regular solution, and x < -10 for the negative solution. The inequality thus becomes x < -10 and x > 4.