Rational Expressions and Functions: math test solutions, Grade 11 – download the PDF

1 Simplifying / 3 pts

1. \(\dfrac{(x+5)(x-3)}{(x-3)(x+3)}=\dfrac{x+5}{x+3}\) (1.5 pt) with \(x\neq 3\) and \(x\neq -3\) (0.5 pt).
2. \(\dfrac{6x^3y}{15xy^2}=\dfrac{2x^2}{5y}\) (0.5 pt) with \(x\neq 0\) and \(y\neq 0\) (0.5 pt).

2 Multiplying and dividing / 4 pts

1. \(\dfrac{(x-5)(x+5)}{x+2}\cdot\dfrac{x(x+2)}{x-5}\) (1 pt) \(=x(x+5)\) (1 pt).
2. \(\dfrac{(x-4)(x+3)}{(x-1)(x+1)}\cdot\dfrac{x+1}{x-4}\) (1 pt) \(=\dfrac{x+3}{x-1}\) (1 pt).

3 Adding and subtracting / 4 pts

1. LCD \((x-3)(x+3)\) (0.5 pt): \(\dfrac{4(x+3)+2(x-3)}{(x-3)(x+3)}\) (0.5 pt) \(=\dfrac{6x+6}{x^2-9}\) (1 pt).
2. LCD \((x-5)(x+1)\) (0.5 pt): \(\dfrac{x(x+1)-3(x-5)}{(x-5)(x+1)}\) (0.5 pt) \(=\dfrac{x^2-2x+15}{(x-5)(x+1)}\) (1 pt).

4 Analyzing a function / 4 pts

1. Vertical asymptote \(x=4\) (1 pt); horizontal asymptote \(y=2\) (1 pt).
2. \(2x+6=0\) gives the \(x\)-intercept \(-3\) (0.5 pt); \(f(0)=\dfrac{6}{-4}=-1.5\) (0.5 pt).
3. \(f(6)=\dfrac{18}{2}=9\) (0.5 pt) and \(f(2)=\dfrac{10}{-2}=-5\) (0.5 pt).

5 Solving an equation / 3 pts

Multiply by \((x-2)(x+2)\): \(3(x+2)+(x^2-4)=12\) (1 pt), so \(x^2+3x-10=0\) and \((x+5)(x-2)=0\) (1 pt).

\(x=2\) makes a denominator zero, so it is extraneous; \(x=-5\) works: \(-\dfrac37+1=\dfrac47=\dfrac{12}{21}\). The only solution is \(x=-5\) (1 pt).

6 Travel time / 2 pts

\(k=rt=45\cdot 4=180\) (1 pt). So \(t=\dfrac{180}{60}=3\): the trip takes 3 hours (1 pt).