Question

chapter 4 - retake ticket

1. find the vertical asymptotes for the rational function: $y = \frac{x-1}{x+3}$
2. find the horizontal asymptote for the rational function: $y = \frac{(x-1)(x-3)}{(x-1)(x-6)}$
3. simplify the rational equation: $y = \frac{4x + 16}{4x + 30}$
4. simplify the rational equation: $\frac{x^2 - 4x - 12}{2x^2 + 4x}$
5. multiply the rational functions: $\frac{7}{x - 4} \cdot \frac{x^2 - 5x + 4}{7x + 14}$
6. divide the rational functions: $\frac{x + 1}{x + 6} \div \frac{x^2 + 5x + 4}{x + 6}$
7. add the rational functions: $\frac{x^2 + 3x + 10}{x^2 + x - 6} + \frac{2}{x + 3}$
8. subtract the rational functions: $\frac{m - 5}{xp} - \frac{m - 1}{xp}$

Explanation:

Problem 1: Find vertical asymptote

Step1: Set denominator to 0

$x+3=0$

Step2: Solve for x

$x=-3$

Problem 2: Find horizontal asymptote

Step1: Cancel common factors

$y=\frac{(x-1)(x-3)}{(x-1)(x-6)}=\frac{x-3}{x-6}$ (for $x
eq1$)

Step2: Compare degrees, find ratio

Degrees of numerator/denominator = 1/1, ratio is $\frac{1}{1}=1$

Problem 3: Simplify rational function

Step1: Factor numerator/denominator

$y=\frac{4(x+4)}{2(2x+15)}$

Step2: Cancel common factor

$y=\frac{2(x+4)}{2x+15}$

Problem 4: Simplify rational function

Step1: Factor numerator/denominator

$\frac{(x-6)(x+2)}{2x(x+2)}$

Step2: Cancel common factor

$\frac{x-6}{2x}$ (for $x
eq-2$)

Problem 5: Multiply rational functions

Step1: Factor numerator/denominator

$\frac{7}{x-4}\cdot\frac{(x-4)(x-1)}{7(x+2)}$

Step2: Cancel common factors

$\frac{x-1}{x+2}$ (for $x
eq4$)

Problem 6: Divide rational functions

Step1: Rewrite as multiply by reciprocal

$\frac{x+1}{x+6}\cdot\frac{x+6}{(x+1)(x+4)}$

Step2: Cancel common factors

$\frac{1}{x+4}$ (for $x
eq-6,-1$)

Problem 7: Add rational functions

Step1: Factor denominator

$\frac{x^2+3x+10}{(x+3)(x-2)}+\frac{2}{x+3}$

Step2: Get common denominator

$\frac{x^2+3x+10}{(x+3)(x-2)}+\frac{2(x-2)}{(x+3)(x-2)}$

Step3: Combine numerators

$\frac{x^2+3x+10+2x-4}{(x+3)(x-2)}$

Step4: Simplify numerator

$\frac{x^2+5x+6}{(x+3)(x-2)}=\frac{(x+3)(x+2)}{(x+3)(x-2)}$

Step5: Cancel common factor

$\frac{x+2}{x-2}$ (for $x
eq-3$)

Problem 8: Subtract rational functions

Step1: Combine over common denominator

$\frac{(m-5)-(m-1)}{xp}$

Step2: Simplify numerator

$\frac{m-5-m+1}{xp}=\frac{-4}{xp}$

Answer:

1. $x=-3$
2. $y=1$
3. $y=\frac{2(x+4)}{2x+15}$
4. $\frac{x-6}{2x}$
5. $\frac{x-1}{x+2}$
6. $\frac{1}{x+4}$
7. $\frac{x+2}{x-2}$
8. $-\frac{4}{xp}$