

4/(2 - x) + 3/(x + 2) >= (3x ^ 2)/(4 - x ^ 2) 

Let's solve the inequality:

\[
\frac{4}{2 - x} + \frac{3}{x + 2} \geq \frac{3x^2}{4 - x^2}
\]

### Step 1: Factorize and analyze the equation
First, recognize that \(4 - x^2 = (2 - x)(2 + x)\), so the right-hand side can be rewritten as:

\[
\frac{3x^2}{(2 - x)(2 + x)}
\]

Thus, the inequality becomes:

\[
\frac{4}{2 - x} + \frac{3}{x + 2} \geq \frac{3x^2}{(2 - x)(2 + x)}
\]

### Step 2: Combine the left-hand side into a single fraction
To combine the fractions on the left-hand side, find a common denominator. The common denominator of \(2 - x\) and \(x + 2\) is \((2 - x)(x + 2)\). Rewriting both fractions:

\[
\frac{4}{2 - x} = \frac{4(x + 2)}{(2 - x)(x + 2)}
\]
\[
\frac{3}{x + 2} = \frac{3(2 - x)}{(2 - x)(x + 2)}
\]

Now, combine them:

\[
\frac{4(x + 2) + 3(2 - x)}{(2 - x)(x + 2)} \geq \frac{3x^2}{(2 - x)(x + 2)}
\]

### Step 3: Simplify the numerator
Simplify the numerator on the left-hand side:

\[
4(x + 2) + 3(2 - x) = 4x + 8 + 6 - 3x = x + 14
\]

So the inequality becomes:

\[
\frac{x + 14}{(2 - x)(x + 2)} \geq \frac{3x^2}{(2 - x)(x + 2)}
\]

### Step 4: Eliminate the denominators
Since the denominators are the same on both sides, we can cancel them (assuming they are not zero). This gives us:

\[
x + 14 \geq 3x^2
\]

### Step 5: Rearrange the terms
Rearrange the terms to form a standard quadratic inequality:

\[
3x^2 - x - 14 \leq 0
\]

### Step 6: Solve the quadratic inequality
Solve the quadratic equation \(3x^2 - x - 14 = 0\) using the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-14)}}{2(3)}
\]
\[
x = \frac{1 \pm \sqrt{1 + 168}}{6}
\]
\[
x = \frac{1 \pm \sqrt{169}}{6}
\]
\[
x = \frac{1 \pm 13}{6}
\]

So, the solutions are:

\[
x = \frac{1 + 13}{6} = \frac{14}{6} = \frac{7}{3}, \quad x = \frac{1 - 13}{6} = \frac{-12}{6} = -2
\]

### Step 7: Analyze the solution intervals
Now, we need to determine where the quadratic expression \(3x^2 - x - 14 \leq 0\). The roots are \(x = \frac{7}{3}\) and \(x = -2\), so the sign of the quadratic changes at these points. We analyze the intervals:

- For \(x < -2\)
- For \(-2 < x < \frac{7}{3}\)
- For \(x > \frac{7}{3}\)

Since we are looking for \(3x^2 - x - 14 \leq 0\), the solution to the inequality lies in the interval \([-2, \frac{7}{3}]\).

### Step 8: Consider domain restrictions
We also need to exclude points where the denominator \( (2 - x)(x + 2) = 0\). These occur at \(x = 2\) and \(x = -2\). Therefore, we must exclude \(x = 2\) and \(x = -2\) from the solution set.

### Final Solution
The solution to the inequality is:

\[
x \in (-2, \frac{7}{3}]
\]

Would you like more details on any of the steps or concepts?

---

### Related Questions:
1. How can you solve quadratic inequalities using a number line?
2. What is the domain of rational expressions?
3. How does the quadratic formula help in solving inequalities?
4. What are some applications of quadratic inequalities in real life?
5. How do you factor quadratic expressions?

### Tip:
Always check for restrictions in the domain when working with rational inequalities, especially when the denominator involves variables.

4/(2 - x) + 3/(x + 2) >= (3x ^ 2)/(4 - x ^ 2)

Learn how to solve the rational inequality 4/(2 - x) + 3/(x + 2) >= (3x^2)/(4 - x^2) using algebraic techniques. This step-by-step guide covers key mathematical concepts like combining fractions, applying the quadratic formula, and understanding domain restrictions. Suitable for students in Grades 10-12.

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