https://file.aiquickdraw.com/mathbot/file/17254771723204t8w85g0.jpg

We are given the inequality:

\[
4 |2w + 3| - 7 \leq 9
\]

### Step-by-Step Solution:

1. **Isolate the absolute value term**:
   \[
   4 |2w + 3| - 7 \leq 9
   \]
   Add 7 to both sides to move the constant:
   \[
   4 |2w + 3| \leq 16
   \]
   
2. **Divide by 4**:
   \[
   |2w + 3| \leq 4
   \]

3. **Solve the absolute value inequality**:
   The inequality \(|2w + 3| \leq 4\) means that \(2w + 3\) is between \(-4\) and \(4\). We can write this as:
   \[
   -4 \leq 2w + 3 \leq 4
   \]

4. **Solve for \(w\)**:
   - Subtract 3 from all parts of the inequality:
     \[
     -4 - 3 \leq 2w \leq 4 - 3
     \]
     \[
     -7 \leq 2w \leq 1
     \]

   - Divide by 2:
     \[
     \frac{-7}{2} \leq w \leq \frac{1}{2}
     \]
     \[
     -\frac{7}{2} \leq w \leq \frac{1}{2}
     \]

### Final Answer:
\[
w \in \left[ -\frac{7}{2}, \frac{1}{2} \right]
\]

Would you like more details on any step or have any specific questions?

### Relative Questions:
1. How do you solve absolute value equations that involve inequalities?
2. What are the properties of absolute values in equations?
3. How do you graph the solution of an absolute value inequality?
4. What are the differences between solving equalities and inequalities?
5. How would you apply this solution in a real-world scenario?

### Tip:
When dealing with absolute values in inequalities, always break the problem into two separate cases: the positive case and the negative case.

Solve the inequality 4|2w + 3| - 7 ≤ 9.

Learn how to solve the absolute value inequality 4|2w + 3| - 7 ≤ 9 with a detailed step-by-step solution. This guide covers algebraic concepts like absolute value inequalities and linear inequalities, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation