## Section 2.7: Polynomial and Rational Inequalities

### Learning Outcomes

• Solve polynomial inequalities algebraically.
• Solve rational inequalities algebraically.

## Solving Polynomial Inequalities

One application of our ability to find intercepts and sketch a graph of polynomials is the ability to solve polynomial inequalities. It is a very common question to ask when a function will be positive and negative.

When we solve an equation and the result is $x=3$, we know there is one solution, which is 3.

When we solve an inequality and the result is $x<3$, we know there are many solutions. We graph the result to better help show all the solutions, and we start with 3.  Three becomes a critical point and then we decide whether to shade to the left or right of it.  The numbers to the right of 3 are larger than 3, so we shade to the right. To solve a polynomial inequality, we must write the inequality with the polynomial on the left and 0 on the right.

Next, we determine the critical points to use to divide the number line into intervals. A critical point is a number which makes the polynomial zero.

We will evaluate the factors of the polynomial. This will identify the interval, or intervals, that contains all the solutions of the polynomial inequality.

We write the solution in interval notation being careful to determine whether the endpoints are included.

### How To: Solve a polynomial inequality

1. Write the inequality with the polynomial on the left and zero on the right
2. Determine the critical points-the points where the polynomial will be zero.
3. Use the critical points to divide the number line into intervals.
4. Test a value in each interval. Indicate which regions are positive and negative.
5. Determine the intervals where the inequality is correct. Write the solution in interval notation.

### Example 1: Solving A Polynomial InequalitY in Factored FORM

Solve the inequality $\left(x+3\right){\left(x+1\right)}^{2}\left(x-4\right)> 0$

### Example 2: Solving A Quadratic InequalitY NOT in Factored FORM

Solve the inequality $6-5t-{t}^{2}\ge 0$

### Example 3: Solving a Polynomial Inequality Not in Factored Form

Solve the inequality ${x}^{4} - 2{x}^{3} - 3{x}^{2} \gt 0$

## Solving Rational Inequalities

In addition to finding when a polynomial function will be positive and negative, we can also find where rational functions are positive and negative.

A rational function is a function that can be written as the quotient of two polynomial functions.  Rational functions will be studied in more detail in the next section.

### A General Note: Rational Function

A rational function is a function that can be written as the quotient of two polynomial functions $P\left(x\right) \text{and} Q\left(x\right)$.

$f\left(x\right)=\dfrac{P\left(x\right)}{Q\left(x\right)}=\dfrac{{a}_{p}{x}^{p}+{a}_{p - 1}{x}^{p - 1}+…+{a}_{1}x+{a}_{0}}{{b}_{q}{x}^{q}+{b}_{q - 1}{x}^{q - 1}+…+{b}_{1}x+{b}_{0}},Q\left(x\right)\ne 0$

### Rational Inequality

A rational inequality is an inequality that contains a rational expression.

Inequalities such as $\frac{3}{2x}>1\text{ , }\frac{2x}{x-3}<4\text{ , }\frac{(2x-3)(x-5)}{x+1)^{2}}<0\text{ , and }\frac{1}{4}-\frac{2}{x^{2}}\leq\frac{3}{x}$ are rational inequalities as they each contain a rational expression.

When we solve a rational inequality, we will use the same techniques we used solving polynomial inequalities. However, we must carefully consider what value might make the rational expression undefined and so must be excluded.

Next, we determine the critical points to use to divide the number line into intervals. A critical point is a number which makes the rational expression zero of undefined.

We will evaluate the factors of the numerator and denominator, and find the quotient in each interval. This will identify the interval, or intervals, that contains all the solutions of the rational inequality.

We write the solution in interval notation being careful to determine whether the endpoints are included.

### How To: Solve a rational inequality

1. Write the inequality as one quotient on the left and zero on the right
2. Determine the critical points-the points where the rational expression will be zero or undefined
3. Use the critical points to divide the number line into intervals.
4. Test a value in each interval. Indicate which regions are positive and negative.
5. Determine the intervals where the inequality is correct. Write the solution in interval notation.

### Example 4: Solving RATIONAL Inequalities in Factored Form

Solve $\dfrac{(x+3)(x-4)}{(x+1)^{2}}\leq 0$

### Example 5: Solving A rational inequalitY not in factored form

Solve $\dfrac{2x^{2}+6x+9}{x+3}>5$

### Try it

Solve $\dfrac{(x-6)(x+1)}{3x^{2}}<0$ and write your answer in interval notation.

## Section 2.7 Homework Exercises

1. Explain critical points and how they are used to solve polynomial and rational inequalities algebraically.

2. Describe the steps needed to solve a rational inequality algebraically.

For each of the following polynomial inequalities, solve and write your answer in interval notation

3. $\text{ }(x-4)(x+3)\lt 0$

4. $\text{ }(x-4)(x+1)\lt 0$

5. $\text{ }(x-1)(3x-4)\geq 0$

6. $\text{ }(x+7)(2x-5)\geq 0$

7. $\text{ }x^{2}+5x+4\lt 0$

8. $\text{ }x^{2}-14x+49\lt 0$

9. $\text{ }(x+8)(x+2)(x-3)\geq 0$

10. $\text{ }(x+5)(x+1)(x-4)\geq 0$

11. $\text{ }(x+8)^{2}(x+5)(x+7)^{3}\gt 0$

12. $\text{ }(x-4)(x+1)^{3}(x-2)^{2}\gt 0$

13. $\text{ }\left(2x^{2}+5x-3\right)\left(x+4\right)\leq 0$

14. $\text{ }\left(3x^{2}-5x-2\right)\left(x+3\right)\leq 0$

15. $\text{ }\left(x^{2}+3x-10\right)\left(x^{2}-1\right)\lt 0$

16. $\text{ }\left(x^{2}+2x-24\right)\left(x^{2}-4\right)\lt 0$

For each of the following rational inequalities, solve and write your answer in interval notation.

17. $\text{ }\dfrac{x-7}{x-1}\lt 0$

18. $\text{ }\dfrac{x+5}{x-4}\lt 0$

19. $\text{ }\dfrac{x+4}{2x-3}\geq 0$

20. $\text{ }\dfrac{x-5}{3x+4}\geq 0$

21. $\text{ }\dfrac{x+32}{x+6}\geq 3$

22. $\text{ }\dfrac{x+68}{x+8}\geq 5$

23. $\text{ }\dfrac{(x+3)(x+5)}{x+2}\geq 0$

24. $\text{ }\dfrac{(x+4)(x-3)}{x+1}\geq 0$

25. $\text{ }\dfrac{(x-2)(x-7)}{3x+2}\leq 0$

26. $\text{ }\dfrac{(x+1)(x+6)}{2x-1}\leq 0$

27. $\text{ }\dfrac{x+6}{x^{2}-5x-24}\leq 0$

28. $\text{ }\dfrac{x+5}{x^{2}-9x+18}\leq 0$

29. $\text{ }\dfrac{(x+7)(x-3)}{(x-5)^{2}}\lt 0$

30. $\text{ }\dfrac{(x+8)(x-1)}{(x+3)^{2}}\lt 0$

31. $\text{ }\dfrac{2x^{2}+6x+7}{x+3}\gt 3$

32. $\text{ }\dfrac{4x^{2}+8x+9}{x+2}\gt 5$