Question

his assignment includes multiple choice and short answer questions. for multiple choice questions, elect the best answer. each is worth 1 mark. marks assigned to short answer questions are ndicated for each question. be sure to show all necessary work.

1. the solution to the rational equation \\(\frac{11}{r} = \frac{7}{5} - \frac{10}{r}\\) is

a. \\(\frac{5}{7}\\)
b. 15
c. 18
d. there is no solution because the variables are eliminated.

1. the solution to the rational equation \\(\frac{3x^2 + 5x - 2}{2x + 4} = -\frac{7}{2}\\) is

a. \\(x = -2\\)
b. \\(x = 2\\)
c. there is no solution because \\(x \
eq -2\\).
d. there is no solution because \\(x \
eq 2\\).

Explanation:

Question 1

Step1: Add $\frac{10}{r}$ to both sides

To solve the equation $\frac{11}{r}=\frac{7}{5}-\frac{10}{r}$, we first add $\frac{10}{r}$ to both sides to get all the $r$-terms on one side.
$$\frac{11}{r}+\frac{10}{r}=\frac{7}{5}-\frac{10}{r}+\frac{10}{r}$$
Simplifying the left side: $\frac{11 + 10}{r}=\frac{21}{r}$, and the right side simplifies to $\frac{7}{5}$. So we have $\frac{21}{r}=\frac{7}{5}$.

Step2: Cross - multiply to solve for $r$

Cross - multiplying gives us $7r=21\times5$.
First, calculate $21\times5 = 105$. Then, $7r = 105$.
Divide both sides by 7: $r=\frac{105}{7}=15$.

Step1: Factor the numerator and denominator

First, factor the numerator $3x^{2}+5x - 2$. We can factor it as $(3x - 1)(x + 2)$ (using the formula $ax^{2}+bx + c=a(x - x_1)(x - x_2)$ where $x_1$ and $x_2$ are the roots). The denominator $2x + 4$ can be factored as $2(x + 2)$. So the equation $\frac{3x^{2}+5x - 2}{2x + 4}=-\frac{7}{2}$ becomes $\frac{(3x - 1)(x + 2)}{2(x + 2)}=-\frac{7}{2}$.

Step2: Note the domain restriction

We must note that the denominator cannot be zero. So $x+2
eq0$, which means $x
eq - 2$.

Step3: Cancel out the common factor (with restriction)

We can cancel out the common factor of $(x + 2)$ from the numerator and the denominator (since $x
eq - 2$), and we get $\frac{3x - 1}{2}=-\frac{7}{2}$.

Step4: Solve for $x$

Multiply both sides by 2: $3x-1=-7$.
Add 1 to both sides: $3x=-7 + 1=-6$.
Divide both sides by 3: $x=-2$. But we know from the domain restriction that $x
eq - 2$. So there is no solution because $x
eq - 2$.

Answer:

B. 15

Question 2