Solving Rational Inequalities

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Introduction to Solving Rational Inequalities

A key approach towards solving rational inequalities depends upon determining the critical values of the rational expression that divides the number line into distinctive open intervals. The critical values are the zeros (0s) of both the numerator as well as the denominator. Note that the zeros of the denominator make the rational expression undefined, so they should be immediately excluded or dismissed as a probable solution. However, zeros of the numerators also require to be verified for its possible inclusion to the entire solution.


Different Ways of How to Solve Rational Inequalities

Struggling with the ways of how to find rational inequalities? Well! There are different ways of solving rational inequalities which are as below:

  • Solving rational inequalities algebraically.

  • Solving inequalities with rational expressions.

  • Solving rational inequalities by graphing.

How to Solve Rational Inequalities

Below are the summarized steps in order to find rational inequalities and solve them.

Step 1: Write the expression of inequality as one quotient on the left and zero (0) on the right.

Step 2: identify the critical points–the points where the rational expression will either be undefined or zero.

Step 3: Use the critical points for dividing the number line into intervals.

Step 4: Test the value of each interval. The number line displays the sign of each factor of the numerator as well as the denominator in each interval. The number line also shows the sign of the quotient.

Step 5: identify the intervals where inequality is appropriate.

Step 6: Write the solution in the form of interval notation.


Rational

This is what a rational expression seems like having a ratio of two polynomials.


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Sometimes we would require solving rational inequalities like these:


Symbol

Words

Example

>

greater than

(x + 1)/(3 − x) > 2

<

less than

x/(x + 7) < −3

greater than or equal to

(x−1)/(5−x) ≥ 0

less than or equal to

(3 − 2x)/(x − 1) ≤ 2


Solved Example

Example: Solve and simplify the given rational inequality:

(x2 - 3x - 4 )/(x2 - 8x + 16) < 0

Solution:

Step 1: Factor out both the numerator and denominator in order to find their zeros. In factored form, we will obtain:

(x+1) (x - 4) / (x - 4) (x - 4) < 0

Step 2: identify the zeros of the rational inequality by establishing each factor equal to zero

Step 3: Solve for x. We get:

Zeros of the numerators: –1 and 4

Zeros of the denominators: 4

Step 4: Consider taking the zeros as critical numbers in order to divide the number line into distinct intervals.

Step 5: Test the validity of each interval by choosing a test value and assessing them into the original rational inequality. Note that the ones in yellow are the numbers we selected.


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If you observe, the only interval providing a true statement is (- 1, 4).

Additionally, the zeros of the numerator don’t check with the original rational inequality so we should disregard them.

Therefore, the final answer is just (−1,4).


Did You Know?

  • When solving a rational inequality, we must first write the inequality with 0 on the right and only one quotient on the left.

  • Critical points are identified for using them to divide the number line into intervals.

  • Once identifying the critical points, factors of the numerator and denominator, and quotient in each interval are found. This will identify the interval, or intervals, which consists of all the solutions of rational inequality.

Conclusion: Solution of rational inequality is written in interval notation. We should be very careful writing the notation as it helps to identify if or not the endpoints are included.

FAQs (Frequently Asked Questions)

Q1. What is a Rational Inequality?

Answer: A rational inequality is an inequality which consists of a rational expression. While solving a rational inequality, we can use distinct methods as we use when solving linear inequalities. We particularly must remember that when we multiply or divide by a negative number, the inequality sign should reverse.


Moreover, when solving a rational inequality, we must carefully consider what value might make rational expression undefined and thus should be excluded.

Q2. What is the Difference Between a Linear Equation and a Rational Inequality?

Answer: When we solve an equation and the outcome is x=7, we are known that there is only one solution, which is 7.


On the other hand, when we solve inequality and the outcome is x > 7, we know that there are many solutions. We graph the outcome solving rational inequalities by graphing in order to better help show all the solutions, and we begin with 7. Seven becomes a critical point and then we plan if to shade to the left or right of it. The numbers to the right of 7 are greater than 7, so we shade to the right.

Q3. What is a Critical Point?

Answer: A critical point is a number that makes the rational expression zero (0) or undefined.

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