Section 4.6 - Solving Polynomial and Rational Inequalities

Section Objectives

1. Solve polynomial inequalities.
2. Solve rational inequalities.

Polynomial Inequalities

Any polynomial inequality can be written in one of these forms

To solve the inequality, we should start by factoring $P(x)$. With knowledge of the polynomial's zeros and end behavior, it is straightforward to solve any of those polynomial inequalities---simply sketch the graph of the polynomial and use it to determine intervals on which the graph is above, below, or on the $x$-axis.

Another approach is to use a sign chart, on which me mark the polynomial's zeros (in order!) and then indicate whether the polynomial's values are positive or negative between the zeros.

Examples

• Solve: $\quad (x-7)(x-2)<0$ (Solution)

• Solve: $\quad x^2 \le -4x$ (Solution)

• Solve: $\quad (x+4)(x+9)(x-5)\ge 0$ (Solution)

• Solve: $\quad (x+1)(x-5)(x^2+9)<0$ (Solution)

• Solve: $\quad x^3+5x^2 \le -4x-20$ (Solution)

Rational Inequalities

Any rational inequality can be written in one of the following forms:

where $P(x)$ and $Q(x)$ are polynomials.

Once an inequality is written in the appropriate form, we can use the same techniques we used to solve polynomial inequalities: use the graph or use a sign chart. We'll use both techniques, but we will focus most of our attention on the sign chart method.

To use the sign chart method...

1. If necessary, use algebra to write the inequality in the appropriate form.
2. Completely factor both the numerator and denominator, and find the zeros of each.
3. Sketch a number line, and indicate all of the zeros you found in step 1. Indicate which are zeros of the numerator and which are zeros of the denominator. You must never include a zero of the denominator in a solution set.
4. By testing values, determine the sign ($+$ or $-$) of the rational function on each interval on your number line.
5. Read the appropriate solution set from your sign chart. Be sure to include the zeros of the numerator if appropriate.

Examples

• Let $\displaystyle f(x)=\frac{-x^2-9x}{x^2+9x+18}$. Solve the inequality $f(x) \ge 0$. (Solution)

• Let $\displaystyle f(x)=\frac{x-1}{x+5}$. Solve $f(x)<0$. (Solution)

• Solve $\displaystyle \frac{-1}{x-6} < \frac{2}{9-x}$. (Solution)

For further help...

1. Do a Google search for "solving polynomial inequalities". You'll find some good videos.
2. Do a Google search for "solving rational inequalities". You'll find some good videos.
3. Use Desmos or Geogebra to graph the inequality. If you enter the inequality, they automatically shade the solution region. This is great for checking your work, but you have to know the ideas!