https://file.aiquickdraw.com/mathbot/file/1727670666438mnfv2kvd.jpg

Let's solve the inequalities shown in the image.

### 1. Inequality: \( 9|x| - 9 \leq 9 \)
- First, isolate the absolute value term:
\[
9|x| - 9 \leq 9
\]
Add 9 to both sides:
\[
9|x| \leq 18
\]
Divide by 9:
\[
|x| \leq 2
\]
This implies:
\[
-2 \leq x \leq 2
\]
The solution is the interval:
\[
[-2, 2]
\]

### 2. Inequality: \( 5|x| - 3 \geq 2 \)
- Isolate the absolute value term:
\[
5|x| - 3 \geq 2
\]
Add 3 to both sides:
\[
5|x| \geq 5
\]
Divide by 5:
\[
|x| \geq 1
\]
This implies two cases:
\[
x \geq 1 \quad \text{or} \quad x \leq -1
\]
So, the solution is:
\[
(-\infty, -1] \cup [1, \infty)
\]

### Interval Notation Summary:
1. For \( 9|x| - 9 \leq 9 \), the interval notation is \( [-2, 2] \).
2. For \( 5|x| - 3 \geq 2 \), the interval notation is \( (-\infty, -1] \cup [1, \infty] \).

Let me know if you have any questions or need further clarification!

Here are 5 related questions for further exploration:
1. How do you solve compound inequalities involving absolute values?
2. What is the difference between strict inequalities (\(<\), \(>\)) and non-strict inequalities (\(\leq\), \(\geq\))?
3. How do we interpret absolute value inequalities geometrically?
4. Can absolute value inequalities have no solution?
5. How do you solve absolute value equations?

**Tip**: When solving absolute value inequalities, always remember to split the inequality into two cases based on the definition of absolute value.

Solve each inequality - graph the solution and give interval notation.

Learn how to solve absolute value inequalities like 9|x| - 9 ≤ 9 and 5|x| - 3 ≥ 2 with step-by-step solutions. Understand how to graph the solution and express it in interval notation. This algebra problem covers key concepts such as absolute value inequalities and their corresponding interval solutions, suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation