Equations and inequalities with Rational Functions

### Rational equations and inequalities

To solve equations involving rational expressions, we have the freedom to clear out fractions before proceeding. After multiplying both sides by the common denominator, we are left with a polynomial equation.

The quadratic formula gives solutions as .

If we look back at the original equation, we notice that there are some numbers that we are not allowed to plug in for . When or , the left-hand side of the equation is not defined due to a division by zero issue. Since neither nor have such an issue, they are both solutions.

### Inequalities

When faced with nonlinear inequalities, such as those involving general rational functions, we make use of a sign chart. The inequality in the following example is not given in factored form, so we have some work to do.

Now that the inequality is in a better form for us to work with, we’ll build a sign chart like we did in the last example.

We see from the chart that will be negative in . At and it is zero. The solution is then: .