The inequality \(|x| \leq a\) can be interpreted as follows:

### Meaning:
1. The absolute value of \(x\) is less than or equal to \(a\).
2. This implies that \(x\) is within a closed interval centered at \(0\) with a radius of \(a\).

### Solution:
The inequality can be written as a compound inequality:
\[
-a \leq x \leq a
\]
This means \(x\) lies between \(-a\) and \(a\), inclusive.

### Examples:
1. If \(a = 3\), the inequality \(|x| \leq 3\) means \(-3 \leq x \leq 3\).
2. If \(a = 0\), the inequality \(|x| \leq 0\) implies \(x = 0\) because the only number whose absolute value is \(0\) is \(0\).

Let me know if you'd like a graphical representation or further examples!

### Questions to Expand Understanding:
1. What happens if the inequality is \(|x| < a\) instead?
2. How does the solution change if \(a\) is negative?
3. Can you derive the interval for \(|x| \leq 5\)?
4. What is the difference between \(|x| \leq a\) and \(|x| > a\)?
5. How does this inequality relate to distance on the number line?

**Tip**: Always remember that \(|x| \leq a\) is valid only when \(a \geq 0\), as absolute values cannot be negative.

Learn how to solve the absolute value inequality |x| ≤ a, including step-by-step guidance on interpreting the inequality and converting it into a compound inequality. This page covers key concepts in algebra and absolute value, suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation