5.2 Rational equations (in class)

If [math]a+b\sqrt{c}[/math] is a solution to the equation [math]\dfrac{3x+1}{x^2-x-12}=\dfrac{1-2x}{x^2-5x+4}[/math] where [math]a[/math] and [math]b[/math] are rational numbers and [math]c[/math] is a prime number, what is the value of [math]c[/math]?

Solve the equation [math]\dfrac{x^2-6x+9}{x^2-4}\cdot\dfrac{x^2+3x+2}{x-3}=0[/math]. If you have two or more answers, write the biggest one.

Solve the equation [math]\dfrac{3}{x^2+10x+12}=-\dfrac{1}{3}[/math]. If you have two or more answers, write the biggest one.

How many positive integer solution does the inequality [math]\dfrac{x-1}{x-2}\leq\dfrac{x-2}{x-1}[/math] have?

If the rational equation [math]\dfrac{2}{x+1}+\dfrac{3}{x^2-k}=0[/math] has only one solution where [math]k[/math] is not an integer, what is the value of [math]k[/math]?

Consider a rational function [math]f(x)=\dfrac{(x+3)(x-2)^2(x+1)^2}{x^3-8}[/math]. Which of the following values is positive?

How many solutions does the equation [math]\displaystyle{\frac{5x+10}{x^2+3x+2}+\frac{7x-28}{x^2-x-12}=2}[/math] have?

How many integer solutions does the inequality [math]\dfrac{(x+2)(x-1)(x+3)}{(x+2)(x^2+4x+5)}\leq0[/math] have?

If [math]x=\dfrac{a}{7}[/math] is not a solution to the inequality [math]\dfrac{(x-1)^2(x-3)}{(x-2)}>0[/math] where [math]a[/math] is an integer, what is the minimum of [math]a[/math]?

How many negative integer solutions does the inequality [math]\dfrac{1}{x^2-1}>\dfrac{4}{x^2-6x+5}[/math] have?