Solving Polynomial and Rational Inequalities

To solve an inequality such as $\frac{P(x)}{Q(x)}>0$ or $\frac{P(x)}{Q(x)}\ge0$ where $P(x)$ and $Q(x)$ are polynomials,

1. Factor $P(x)$ and $Q(x)$ completely over the real numbers.

2. Mark the zeros of $P(x)$ and $Q(x)$ on a number line.

3. Determine the sign of $\frac{P(x)}{Q(x)}$ on each of the resulting intervals.

4. Select the intervals corresponding to the sign of the original inequality. (If the inequality is not a strict inequality, include the zeros of $P(x)$ in the solution.)

In determining the sign of $\frac{P(x)}{Q(x)}$ on each interval, we can use the following:

If $(x-c)^n$ is the highest power of $x-c$ which is a factor of $P(x)$ or $Q(x)$, then

A. the sign of $\frac{P(x)}{Q(x)}$ changes at $c$ if $n$ is odd; and

B. the sign of $\frac{P(x)}{Q(x)}$ does not change at $c$ if $n$ is even.
Ex 1 Solve the inequality $x^2-2x-3>0$.

Sol Factoring gives $(x-3)(x+1)>0$; so marking off 3 and -1 on a number line and using the facts that $x=0\Rightarrow y=-3$ and that the exponents of $x-3$ and $x+1$ are both odd, we get the sign chart shown below:

Therefore the solution is given by $(-\infty,-1)\cup(3,\infty)$.

Ex 2 Solve the inequality

$$\frac{x^2-2x-24}{x^2-8x-20}\ge0.$$

Sol Factoring gives

$$\frac{(x-6)(x+4)}{(x-10)(x+2)}\ge0.$$

Marking off 6,-4,10, and -2 on a number line and using the facts that $x=0\Rightarrow y=24/20=6/5>0$ and that the exponents of all the factors are odd, we get the sign chart shown below:

Since the inequality is not strict, we can include the zeros of the numerator; so the solution is given by $(-\infty,-4]\cup(-2,6]\cup(10,\infty)$.

Pr 1 Solve the inequality $x^3-x^2-6x>0$.

Pr 2 Solve the inequality $x^4-x^3-6x^2<0$.

Pr 3 Solve the inequality

$$\frac{x^2-5x+4}{x^2-4}\le0.$$

Pr 4 Solve the inequality

$$\frac{2x^2-x-10}{12+x-x^2}>0.$$

Pr 5 Solve the inequality

$$\frac{x^4-10x^2+9}{x^4-16}\ge0.$$

Pr 6 Solve the inequality

$$\frac{2x^3-6x}{(x^2+1)^3}>0.$$

Pr 7 Solve the inequality

$$\frac{2x^4-5x^3+3x^2}{(x^2-8x+16)(3x^2+5x-2)}\ge0.$$

Pr 8 Find all values of $x$ for which $x<\frac{16}{x}$.

Pr 9 Find all values of $x$ for which

$$\frac{x}{x-4}<\frac{x-5}{x+1}.$$

Go to Solutions.

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