Problem 70 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution of the inequality \(|x|>5\) is \(x \in (-\infty, -5)\) or \(x \in (5, \infty)\).

Step 1: Define the Inequality

First define the inequality without absolute value: \(|x|>5\) which can also be written as \(x > 5\) or \(x < -5\)

Step 2: Solve for x in both cases

Starting with the positive scenario: \(x > 5\). This indicates that x can be any value greater than 5. For the negative scenario: \(x < -5\). Here, x can be any value less than -5.

Step 3: Conclusion

Combine the solutions from both the positive and negative scenarios. This concludes that x can either be less than -5 or greater than 5. Therefore, the set of all possible solutions will be \(x \in (-\infty, -5)\) or \(x \in (5, \infty)\). This is an absolute value inequality and will have two solutions