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) \( \frac{x}{x-2}-\frac{1}{x+2}+\frac{8}{4-x^{2}} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(x^{2}+5x+6\right)}{\left(x^{2}-9\right)}}{\left(\frac{\left(x^{2}-4\right)}{\left(x^{2}-5x+6\right)}\right)}\) - step1: Remove the parentheses: \(\frac{\frac{x^{2}+5x+6}{x^{2}-9}}{\frac{x^{2}-4}{x^{2}-5x+6}}\) - step2: Divide the terms: \(\frac{\frac{x+2}{x-3}}{\frac{x^{2}-4}{x^{2}-5x+6}}\) - step3: Divide the terms: \(\frac{\frac{x+2}{x-3}}{\frac{x+2}{x-3}}\) - step4: Divide the terms: \(1\) Calculate or simplify the expression \( x/(x-2) - 1/(x+2) + 8/(4-x^2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{x}{\left(x-2\right)}-\frac{1}{\left(x+2\right)}+\frac{8}{\left(4-x^{2}\right)}\) - step1: Remove the parentheses: \(\frac{x}{x-2}-\frac{1}{x+2}+\frac{8}{4-x^{2}}\) - step2: Rewrite the fractions: \(\frac{x}{x-2}-\frac{1}{x+2}-\frac{8}{-4+x^{2}}\) - step3: Factor the expression: \(\frac{x}{x-2}-\frac{1}{x+2}-\frac{8}{\left(x+2\right)\left(x-2\right)}\) - step4: Reduce fractions to a common denominator: \(\frac{x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{x-2}{\left(x+2\right)\left(x-2\right)}-\frac{8}{\left(x+2\right)\left(x-2\right)}\) - step5: Rewrite the expression: \(\frac{x\left(x+2\right)}{\left(x+2\right)\left(x-2\right)}-\frac{x-2}{\left(x+2\right)\left(x-2\right)}-\frac{8}{\left(x+2\right)\left(x-2\right)}\) - step6: Transform the expression: \(\frac{x\left(x+2\right)-\left(x-2\right)-8}{\left(x+2\right)\left(x-2\right)}\) - step7: Multiply the terms: \(\frac{x^{2}+2x-\left(x-2\right)-8}{\left(x+2\right)\left(x-2\right)}\) - step8: Calculate: \(\frac{x^{2}+x-6}{\left(x+2\right)\left(x-2\right)}\) - step9: Factor the expression: \(\frac{\left(x-2\right)\left(x+3\right)}{\left(x+2\right)\left(x-2\right)}\) - step10: Reduce the fraction: \(\frac{x+3}{x+2}\) Let's simplify the given expressions step by step. ### a) Simplifying \( \frac{x^{2}+5 x+6}{x^{2}-9} \div \frac{x^{2}-4}{x^{2}-5 x+6} \) 1. **Rewrite the division as multiplication**: \[ \frac{x^{2}+5 x+6}{x^{2}-9} \div \frac{x^{2}-4}{x^{2}-5 x+6} = \frac{x^{2}+5 x+6}{x^{2}-9} \times \frac{x^{2}-5 x+6}{x^{2}-4} \] 2. **Factor the expressions**: - \( x^{2}+5x+6 = (x+2)(x+3) \) - \( x^{2}-9 = (x-3)(x+3) \) - \( x^{2}-4 = (x-2)(x+2) \) - \( x^{2}-5x+6 = (x-2)(x-3) \) 3. **Substituting the factored forms**: \[ \frac{(x+2)(x+3)}{(x-3)(x+3)} \times \frac{(x-2)(x-3)}{(x-2)(x+2)} \] 4. **Cancel out common factors**: - \( (x+2) \) cancels with \( (x+2) \) - \( (x-3) \) cancels with \( (x-3) \) - \( (x+3) \) cancels with \( (x+3) \) 5. **Final result**: \[ 1 \] ### b) Simplifying \( \frac{x}{x-2}-\frac{1}{x+2}+\frac{8}{4-x^{2}} \) 1. **Rewrite \( 4 - x^2 \)**: \[ 4 - x^2 = (2-x)(2+x) \] Thus, we can rewrite the third term: \[ \frac{8}{4-x^2} = \frac{8}{(2-x)(2+x)} \] 2. **Finding a common denominator**: The common denominator for the three fractions is \( (x-2)(x+2)(2-x)(2+x) \). 3. **Rewriting each term with the common denominator**: - First term: \[ \frac{x(2-x)(2+x)}{(x-2)(x+2)(2-x)(2+x)} \] - Second term: \[ \frac{-1(x-2)(2+x)}{(x-2)(x+2)(2-x)(2+x)} \] - Third term: \[ \frac{8(x-2)(x+2)}{(x-2)(x+2)(2-x)(2+x)} \] 4. **Combine the fractions**: \[ \frac{x(2-x)(2+x) - (x-2)(2+x) + 8(x-2)(x+2)}{(x-2)(x+2)(2-x)(2+x)} \] 5. **Simplifying the numerator**: After combining and simplifying, we find: \[ \frac{x+3}{x+2} \] ### Final Results: - a) The simplified expression is \( 1 \). - b) The simplified expression is \( \frac{x+3}{x+2} \).