For what value of 𝑟 does this equation, two 𝑥 plus five over 𝑥 minus three equals 𝑟, have no solutions?
The place where 𝑟 would have no solutions would be the horizontal asymptote of this function. To find the horizontal asymptotes of a rational function, we have to consider the degree of the polynomials in the numerator and the denominator. If both polynomials are the same degree, divide the coefficient of the highest-degree terms.
That definition has a lot of vocabulary words, so let’s walk slowly through it. First, it says if both polynomials — that means the polynomial in the numerator and the polynomial in the denominator — have the same degree; the degree of the polynomial is the highest degree of all of the terms. Our numerator has a degree of one because we only have 𝑥 to the first power. Our denominator also has a degree of one. The numerator and the denominator here have the same degree.
Because that’s true, our definition tells us divide the coefficient of the highest-degree terms. In the numerator, two is the highest coefficient of the degree of terms, and in the denominator, one is the highest coefficient. All we need to say to find the asymptote, to find the place, where there’s no solution is divide two by one. When 𝑟 equals two, this equation has no solutions.