Solving Rational Inequalities

As with solving polynomial inequalities, the first step to solving rational inequalities is to find the zeros.  Because a rational expression consists of the ratio of two polynomials, the zeroes for both polynomials will be needed.

The zeros in the numerator are $x$-values at which the rational inequality crosses from negative to positive or from positive to negative. The zeros in the denominator are $x$-values are at which the rational inequality is undefined, the result of dividing by zero.

Example

Consider the rational inequality:

$\frac{x^2+2x-3}{x^2-4}>0$

This equation can be factored to give:

$\frac{(x+3)(x-1)}{(x+2)(x-2)}\geq 0$

The numerator has zeros at $x=-3$ and $x=1$. The denominator has zeros at $x=-2$ and $x=2$.

As $x$ crosses rightward past $-3$, $(x+3)$ becomes positive. At that same point, $(x-1)$, $(x+2)$, and $(x-2)$ are all negative. The product of a positive and three negatives is negative, so the rational expression becomes negative as it crosses $x=-3$ in the rightward direction.

The same process can be used to determine that the rational expression is positive after passing the zero at $x=-2$, is negative after passing $x=1$, and is positive after passing $x=2$.

Thus we can conclude that for $x$ values on the open interval from $-\infty$ to $-3$, the rational expression is negative. From $-3$ to $-2$, it is positive; from $-2$ to $1$ it is negative; from $1$ to $2$ it is positive, and from $2$ to $\infty$ it is negative.

Because the inequality is written as $\geq0$ as opposed to $>0$, we will need to evaluate the $x$ values at zeros to determine whether the function is defined.

In the case of $x=-2$ and $x=2$, the rational function has a denominator equal to zero and becomes undefined.

In the case of $x=-3$ and $x=1$, the rational function has a numerator equal to zero, which makes the function overall equal to zero, making it inclusive in the solution.

Thus, the full solution is:

[-3, -2), [1, 2) 

Graph of example

Graph of a rational polynomial with the equation $y=\frac{x^2+2x-3}{x^2-4}$. For $x$ values that are zeros for the numerator polynomial, the rational function overall is equal to zero. For $x$ values that are zeros for the denominator polynomial, the rational function is undefined, with a vertical asymptote forming instead.