Activity 10 Simplify the following $$ \frac{ ext { expressions }}{ ext { a) } \frac{x^{2}+5 x+6}{x^{2}-9} \div \frac{x^{2}-4}{x^{2}-5 x+6}} $$ b) $$ \frac{x}{x-2}-\frac{1}{x+2}+\frac{8}{4-x^{2}} $$

Question

Activity 10 Simplify the following $$ \frac{	ext { expressions }}{	ext { a) } \frac{x^{2}+5 x+6}{x^{2}-9} \div \frac{x^{2}-4}{x^{2}-5 x+6}} $$ b) $$ \frac{x}{x-2}-\frac{1}{x+2}+\frac{8}{4-x^{2}} $$

UpStudy AI · Accepted Answer

Let's simplify each expression step by step.

---

### **a) Simplify**
$$
\frac{x^{2} + 5x + 6}{x^{2} - 9} \div \frac{x^{2} - 4}{x^{2} - 5x + 6}
$$

**Step 1: Rewrite the Division as Multiplication**
$$
\frac{x^{2} + 5x + 6}{x^{2} - 9} \div \frac{x^{2} - 4}{x^{2} - 5x + 6} = \frac{x^{2} + 5x + 6}{x^{2} - 9} \times \frac{x^{2} - 5x + 6}{x^{2} - 4}
$$

**Step 2: Factor All Quadratic Expressions**
$$
\begin{align*}
x^{2} + 5x + 6 &= (x + 2)(x + 3) \
x^{2} - 9 &= (x + 3)(x - 3) \
x^{2} - 5x + 6 &= (x - 2)(x - 3) \
x^{2} - 4 &= (x + 2)(x - 2)
\end{align*}
$$

**Step 3: Substitute the Factored Forms**
$$
\frac{(x + 2)(x + 3)}{(x + 3)(x - 3)} \times \frac{(x - 2)(x - 3)}{(x + 2)(x - 2)}
$$

**Step 4: Cancel Out Common Factors**
- $$x + 3$$ cancels out.
- $$x + 2$$ cancels out.
- $$x - 2$$ cancels out.
- $$x - 3$$ cancels out.

**Step 5: Simplify**
$$
1
$$

**Final Answer for Part a:**
$$
1
$$

---

### **b) Simplify**
$$
\frac{x}{x - 2} - \frac{1}{x + 2} + \frac{8}{4 - x^{2}}
$$

**Step 1: Factor the Denominator $$4 - x^{2}$$**
$$
4 - x^{2} = (2)^2 - x^{2} = (2 - x)(2 + x) = -(x - 2)(x + 2)
$$
So,
$$
\frac{8}{4 - x^{2}} = \frac{8}{-(x - 2)(x + 2)} = -\frac{8}{(x - 2)(x + 2)}
$$

**Step 2: Rewrite the Expression with a Common Denominator**
The common denominator is $$(x - 2)(x + 2)$$.

$$
\frac{x}{x - 2} = \frac{x(x + 2)}{(x - 2)(x + 2)}
$$
$$
-\frac{1}{x + 2} = -\frac{(x - 2)}{(x - 2)(x + 2)}
$$
$$
-\frac{8}{(x - 2)(x + 2)} = -\frac{8}{(x - 2)(x + 2)}
$$

**Step 3: Combine the Fractions**
$$
\frac{x(x + 2) - (x - 2) - 8}{(x - 2)(x + 2)}
$$

**Step 4: Expand and Simplify the Numerator**
$$
x(x + 2) - (x - 2) - 8 = x^2 + 2x - x + 2 - 8 = x^2 + x - 6
$$

**Step 5: Factor the Numerator**
$$
x^2 + x - 6 = (x - 2)(x + 3)
$$

**Step 6: Substitute Back and Cancel Common Factors**
$$
\frac{(x - 2)(x + 3)}{(x - 2)(x + 2)} = \frac{x + 3}{x + 2} \quad \text{(for } x \neq 2 \text{)}
$$

**Final Answer for Part b:**
$$
\frac{x + 3}{x + 2}
$$

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**Summary of Solutions:**

- **a)** The expression simplifies to **$$1$$**.
- **b)** The expression simplifies to **$$\dfrac{x + 3}{x + 2}$$**.
a) $$1$$ b) $$\dfrac{x + 3}{x + 2}$$

Activity 10 Simplify the following \( \frac{\text { expressions }}{\text { a) } \frac{x^{2}+5 x+6}{x^{2}-9} \div \frac{x^{2}-4}{x^{2}-5 x+6}} \) b) \( \frac{x}{x-2}-\frac{1}{x+2}+\frac{8}{4-x^{2}} \)

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Solución

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