Solve this Rational Inequality (x + 12)/(x + 2) ≥ 2

We are tasked with solving the rational inequality:

\[
\frac{x + 12}{x + 2} \geq 2
\]

### Step 1: Rearrange the inequality

First, subtract 2 from both sides to bring all terms to one side:

\[
\frac{x + 12}{x + 2} - 2 \geq 0
\]

Next, express \( 2 \) with the same denominator as \( \frac{x + 12}{x + 2} \):

\[
\frac{x + 12}{x + 2} - \frac{2(x + 2)}{x + 2} \geq 0
\]

Simplify the right-hand side:

\[
\frac{x + 12 - 2(x + 2)}{x + 2} \geq 0
\]

Distribute and simplify the numerator:

\[
\frac{x + 12 - 2x - 4}{x + 2} \geq 0
\]

\[
\frac{-x + 8}{x + 2} \geq 0
\]

### Step 2: Find critical points

The inequality \(\frac{-x + 8}{x + 2} \geq 0\) changes sign at the points where the numerator or denominator equals zero.

1. Set the numerator \(-x + 8 = 0\):

   \[
   x = 8
   \]

2. Set the denominator \(x + 2 = 0\):

   \[
   x = -2
   \]

So the critical points are \( x = 8 \) and \( x = -2 \). These values divide the real number line into three intervals: \((- \infty, -2)\), \((-2, 8)\), and \((8, \infty)\).

### Step 3: Test intervals

We now test points from each interval in the expression \(\frac{-x + 8}{x + 2}\):

1. **Interval \((- \infty, -2)\)**: Choose \( x = -3 \):

   \[
   \frac{-(-3) + 8}{-3 + 2} = \frac{3 + 8}{-1} = \frac{11}{-1} = -11
   \]
   Since \(-11\) is less than zero, this interval does not satisfy the inequality.

2. **Interval \((-2, 8)\)**: Choose \( x = 0 \):

   \[
   \frac{-0 + 8}{0 + 2} = \frac{8}{2} = 4
   \]
   Since 4 is greater than zero, this interval satisfies the inequality.

3. **Interval \((8, \infty)\)**: Choose \( x = 9 \):

   \[
   \frac{-9 + 8}{9 + 2} = \frac{-1}{11} = -\frac{1}{11}
   \]
   Since \(-\frac{1}{11}\) is less than zero, this interval does not satisfy the inequality.

### Step 4: Consider boundary points

- At \( x = 8 \), the expression becomes zero:
  \[
  \frac{-8 + 8}{8 + 2} = 0
  \]
  Since we are looking for values where the expression is greater than or equal to zero, \( x = 8 \) is included in the solution.

- At \( x = -2 \), the expression is undefined because the denominator is zero. So, \( x = -2 \) is not included in the solution.

### Final Solution

The solution to the inequality is:

\[
x \in (-2, 8]
\]

---

Do you need further clarification or details on any of these steps?

### Related Questions:
1. How do you solve other rational inequalities with different denominators?
2. What is the difference between strict inequalities and non-strict inequalities?
3. How do asymptotes affect the solution of rational inequalities?
4. How can we check for extraneous solutions in inequalities?
5. How do we handle rational inequalities involving higher degree polynomials?

### Tip:
Always check the critical points by plugging them back into the original inequality to avoid including undefined values or missing valid boundary solutions.

Master how to solve the rational inequality (x + 12)/(x + 2) ≥ 2 using critical points and sign testing. This detailed step-by-step solution is ideal for high school students in Grades 10-12, covering algebraic concepts and inequalities.

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