5.4 Rational expression concepts -- rational equations

Rational equations

A rational equation means that you are setting two rational expressions equal to each other. The goal is to solve for x; that is, find the x value(s) that make the equation true.

Suppose I told you that:

If you think about it, the x in this equation has to be a 3. That is to say, if x=3 then this equation is true; for any other x value, this equation is false.

This leads us to a very general rule.

A very general rule about rational equations

If you have a rational equation where the denominators are the same, then the numerators must be the same.

This in turn suggests a strategy: find a common denominator, and then set the numerators equal.

As always, it is vital to remember what we have found here. We started with the equation $\frac{3}{x^2+\text{12}x+\text{36}}=\frac{\mathrm{4x}}{x^3+{\mathrm{4x}}^2-\text{12}x}$ . We have concluded now that if you plug $x=–30$ into that equation, you will get a true equation (you can verify this on your calculator). For any other value, this equation will evaluate false.

To put it another way: if you graphed the functions $\frac{3}{x^2+\text{12}x+\text{36}}$ and $\frac{\mathrm{4x}}{x^3+{\mathrm{4x}}^2-\text{12}x}$ , the two graphs would intersect at one point only: the point when $x=–30$ .