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) \( \frac{x}{r-2}-\frac{1}{x+2}+\frac{8}{4-x^{2}} \)
Real Tutor Solution
Answer
Solution
Reviewed and approved by the UpStudy tutoring team
Explain 
Simplify this solution The Deep Dive
To simplify the expression \( \frac{x^{2}+5 x+6}{x^{2}-9} \div \frac{x^{2}-4}{x^{2}-5 x+6} \), first, rewrite the division as multiplication: \[ \frac{x^{2}+5 x+6}{x^{2}-9} \times \frac{x^{2}-5 x+6}{x^{2}-4} \] Factor each polynomial: - \( x^{2}+5 x+6 = (x+2)(x+3) \) - \( x^{2}-9 = (x-3)(x+3) \) - \( x^{2}-4 = (x-2)(x+2) \) - \( x^{2}-5x+6 = (x-2)(x-3) \) Putting the factored forms into the expression gives: \[ \frac{(x+2)(x+3)}{(x-3)(x+3)} \times \frac{(x-2)(x-3)}{(x-2)(x+2)} \] Now, cancel like terms: - The \( (x+2) \) terms cancel. - The \( (x-2) \) terms cancel. - The \( (x+3) \) terms cancel. This simplifies to: \[ \frac{1}{(x-3)} \] For part b, start with the expression: \[ \frac{x}{r-2}-\frac{1}{x+2}+\frac{8}{4-x^{2}} \] First, factor the last term as \( 4-x^{2} = (2-x)(2+x) \), and rewrite the denominator for consistency. The expression becomes: \[ \frac{x}{r-2}-\frac{1}{x+2}+\frac{8}{(2-x)(2+x)} \] To simplify this, you will need a common denominator, which can be \( (r-2)(x+2)(2-x)(2+x) \). Rewrite each term with this common denominator and combine them. This process involves some algebraic manipulation, but be careful to add fractions correctly to avoid common mistakes like ignoring the negative sign when dealing with \( (2-x) \). Ultimately, you will combine and simplify the fractions to arrive at your final result.
