Rational equations and inequalities

1003136401

Level: 
A
Choose the operation, which most effectively eliminates fractions from the equation. \[ 3+\frac2{x+4}=\frac1{3x+12} \]
multiplying both sides by \( 3x+12 \)
multiplying both sides by \( (x+4)(3x+12) \)
subtracting \( \frac2{x+4} \) from both sides
multiplying both sides by \( 12x \)

1003136402

Level: 
A
Choose the operation, which most effectively eliminates fractions from the equation. \[ \frac2{x^2-9}+\frac3{3-x}=\frac{x+1}{2x} \]
multiplying both sides by \( 2x\left(x^2-9\right) \)
multiplying both sides by \( 2x\left(x^2-9\right)(3-x) \)
multiplying both sides by \( 2x^2-9 \)
multiplying both sides by \( 18x^2 \)

1003136403

Level: 
A
Choose the operation, which most effectively eliminates fractions from the equation. \[ \frac{2x}{x^2-25}+\frac{3+x}{5-x}=\frac{x+1}{x+5} \]
multiplying both sides by \( x^2-25 \)
multiplying both sides by \( (5-x)\left(x^2-25\right) \)
multiplying both sides by \( x^2+25 \)
multiplying both sides by \( (5-x)(x+5)\left(x^2-25\right) \)

1003136405

Level: 
A
Choose the resulting form of the given equation after multiplying both sides by \( x^2-25 \). \[ 1+\frac x{5-x}=\frac{3+x}{x+5}+\frac x{x^2-25} \]
\( x^2-25-x(x+5)=(3+x)(x-5)+x \)
\( x^2-25+x(x+5)=(3+x)(x-5)+x \)
\( x^2-25-x(x-5)=(3+x)(x-5)+x \)
\( x^2-25+x(x-5)=(3+x)(x+5)+x \)

1003136406

Level: 
A
Choose the resulting form of the given equation after multiplying both sides by \( x^2+5x+6 \). \[ -1+\frac{x-2}{x+2}=\frac{1+x}{x^2+5x+6}-\frac x{x+3} \]
\( -x^2-5x-6+(x-2)(x+3)=1+x-x(x+2) \)
\( -1\left(x^2+5x+6\right)+(x-2)(x-3)=1+x-x(x-2) \)
\( -1\left(x^2+5x+6\right)+(x-2)(x+3)=1+x+x(x+2) \)
\( -x^2-5x-6+(x-2)(x+2)=1+x-x(x+3) \)

1103044801

Level: 
A
Given graphs of the functions \( f(x) =2x^2-2x-4 \) and \( g(x) = 2x+2 \), find the domain of the equation\( \frac{2x^2-2x-4}{2x+2} = 10 \).
\( \mathbb{R}\setminus\{-1\} \)
\( \mathbb{R}\setminus\{-1;2\} \)
\( \mathbb{R}\setminus\{-1;2;3\} \)
\( \mathbb{R}\setminus\{-1;3\} \)

1103044802

Level: 
A
Given graphs of the functions \( f(x)=x^2-4x \) and \( g(x) = 4x^2-16x+12 \), find the domain of the equation \( \frac{4x^2-16x+12}{x^2-4x}=6 \).
\( \mathbb{R}\setminus\{0;4\} \)
\( \mathbb{R}\setminus\{1;3\} \)
\( \mathbb{R}\setminus\{0;1;3;4\} \)
\( \mathbb{R}\setminus\{2\} \)