To solve the inequality \( \frac{x + 12}{x + 2} \geq 2 \), let's break it down step by step:

### Step 1: Move all terms to one side
We subtract 2 from both sides to get a single expression:
\[
\frac{x + 12}{x + 2} - 2 \geq 0
\]
Now, we combine the terms on the left-hand side under a common denominator:
\[
\frac{x + 12}{x + 2} - 2 = \frac{x + 12 - 2(x + 2)}{x + 2} = \frac{x + 12 - 2x - 4}{x + 2} = \frac{-x + 8}{x + 2}
\]
Thus, the inequality becomes:
\[
\frac{-x + 8}{x + 2} \geq 0
\]

### Step 2: Solve the inequality
We now need to determine when the expression \( \frac{-x + 8}{x + 2} \geq 0 \).

This inequality holds when:

1. The numerator \( -x + 8 \geq 0 \), or equivalently \( x \leq 8 \).
2. The denominator \( x + 2 > 0 \), or equivalently \( x > -2 \) (since the denominator cannot be zero or negative for the fraction to be defined and non-negative).

### Step 3: Find the solution set
Now we combine the conditions:
- From \( -x + 8 \geq 0 \), we have \( x \leq 8 \).
- From \( x + 2 > 0 \), we have \( x > -2 \).

Thus, the solution for the inequality is:
\[
-2 < x \leq 8
\]

### Final Answer:
The solution to the inequality is:
\[
x \in (-2, 8]
\]

Would you like more details or have any questions?

### Related questions:
1. How do you solve inequalities involving absolute values?
2. Can you solve rational inequalities with quadratic expressions in the numerator?
3. How does the method differ when solving strict inequalities (i.e., \(>\) instead of \( \geq \))?
4. What happens when the denominator of a rational inequality includes variables that can make it zero?
5. How can inequalities be graphed on a number line?

### Tip:
When solving rational inequalities, always pay attention to the values that make the denominator zero. These values often define boundary points in the solution set.

Learn how to solve the rational inequality (x + 12) / (x + 2) ≥ 2 with a detailed step-by-step solution. This problem covers key algebraic concepts, including solving inequalities, combining terms, and analyzing critical points. 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