https://file.aiquickdraw.com/mathbot/file/1729222010021ahgaikn7.jpg

Let's solve the problem step by step.

**Given:**
\[
y = 6 - \left|\frac{x}{2} + 2\right|
\]
and we need to find the values of \(x\) such that \(y\) is at most 4.

**Step 1:** Set up the inequality for \(y\):
\[
6 - \left|\frac{x}{2} + 2\right| \leq 4
\]

**Step 2:** Rearrange the inequality:
\[
-\left|\frac{x}{2} + 2\right| \leq -2
\]

By multiplying both sides by -1 (and flipping the inequality sign), we get:
\[
\left|\frac{x}{2} + 2\right| \geq 2
\]

**Step 3:** Solve the absolute value inequality:
\[
\frac{x}{2} + 2 \geq 2 \quad \text{or} \quad \frac{x}{2} + 2 \leq -2
\]

**Case 1:** Solve \(\frac{x}{2} + 2 \geq 2\):
\[
\frac{x}{2} \geq 0 \implies x \geq 0
\]

**Case 2:** Solve \(\frac{x}{2} + 2 \leq -2\):
\[
\frac{x}{2} \leq -4 \implies x \leq -8
\]

**Step 4:** Write the solution in interval notation:
The values of \(x\) that satisfy the given conditions are \(x \leq -8\) or \(x \geq 0\).

Therefore, the solution in interval notation is:
\[
(-\infty, -8] \cup [0, \infty)
\]

Would you like further details or have any questions about the solution?

### Related Questions:
1. How would the solution change if \(y\) was at least 4 instead of at most 4?
2. How do you solve absolute value inequalities when the inequality symbol is reversed (e.g., \(|x| < a\))?
3. What is the graphical interpretation of this solution on a number line?
4. How does shifting the expression inside the absolute value (e.g., \(x/2 + 2\)) affect the solution?
5. How would the solution change if we used a different absolute value equation, such as \(6 - |x + 3|\)?

### Tip:
When solving absolute value inequalities, always consider splitting the problem into two cases: one for the positive and one for the negative scenario of the expression inside the absolute value.

Use interval notation to represent all values of x satisfying the given conditions: y = 6 - |x/2 + 2|, and y is at most 4.

Learn how to solve the inequality y = 6 - |x/2 + 2| with the condition y ≤ 4. This step-by-step solution covers absolute value properties and demonstrates how to express the solution in interval notation. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation