Solve The Rational Inequality $\frac{5}{x+3}\ \textless \ 5$. Express The Answer In Interval Form.A. $(-\infty, -3) \cup (2, \infty$\] B. $(-\infty, -2) \cup (3, \infty$\] C. $(-\infty, -3) \cup (-2, \infty$\] D.
=====================================================
Introduction
Rational inequalities are a type of mathematical inequality that involves a rational expression, which is a fraction of two polynomials. Solving rational inequalities requires a different approach than solving linear or quadratic inequalities. In this article, we will focus on solving the rational inequality and express the answer in interval form.
Understanding Rational Inequalities
A rational inequality is an inequality that involves a rational expression. The rational expression is a fraction of two polynomials, where the numerator and denominator are both polynomials. Rational inequalities can be solved using various methods, including factoring, the sign chart method, and the use of a graphing calculator.
Solving the Rational Inequality
To solve the rational inequality , we need to follow these steps:
Step 1: Multiply Both Sides by the Denominator
The first step in solving the rational inequality is to multiply both sides by the denominator, which is . This will eliminate the fraction and make it easier to solve the inequality.
Step 2: Distribute the 5
Next, we need to distribute the 5 to the terms inside the parentheses.
Step 3: Subtract 15 from Both Sides
Now, we need to subtract 15 from both sides to isolate the term with the variable.
Step 4: Divide Both Sides by 5
Finally, we need to divide both sides by 5 to solve for x.
Expressing the Answer in Interval Form
The solution to the rational inequality is . However, we need to express the answer in interval form. The interval form of the solution is .
Conclusion
Solving rational inequalities requires a different approach than solving linear or quadratic inequalities. By following the steps outlined in this article, we can solve the rational inequality and express the answer in interval form.
Frequently Asked Questions
Q: What is a rational inequality?
A: A rational inequality is an inequality that involves a rational expression, which is a fraction of two polynomials.
Q: How do I solve a rational inequality?
A: To solve a rational inequality, you need to follow the steps outlined in this article, including multiplying both sides by the denominator, distributing the terms, subtracting the constant term, and dividing both sides by the coefficient of the variable.
Q: What is the interval form of the solution?
A: The interval form of the solution is .
References
- [1] "Rational Inequalities" by Math Open Reference
- [2] "Solving Rational Inequalities" by Khan Academy
Discussion
The solution to the rational inequality is . However, some of you may be wondering why the solution is not . The reason is that the denominator of the rational expression is , not . Therefore, the solution to the rational inequality is , not .
I hope this article has helped you understand how to solve rational inequalities and express the answer in interval form. If you have any questions or need further clarification, please don't hesitate to ask.
Answer
A. is incorrect
B. is correct
C. is incorrect
D. is not a valid option
=====================================================
Introduction
Rational inequalities can be a challenging topic for many students. In our previous article, we discussed how to solve rational inequalities and express the answer in interval form. However, we know that there are many questions and concerns that students may have when it comes to rational inequalities. In this article, we will address some of the most frequently asked questions about rational inequalities.
Q&A
Q: What is a rational inequality?
A: A rational inequality is an inequality that involves a rational expression, which is a fraction of two polynomials.
Q: How do I know if a rational inequality is true or false?
A: To determine if a rational inequality is true or false, you need to follow the steps outlined in our previous article, including multiplying both sides by the denominator, distributing the terms, subtracting the constant term, and dividing both sides by the coefficient of the variable.
Q: What is the difference between a rational inequality and a rational equation?
A: A rational equation is an equation that involves a rational expression, whereas a rational inequality is an inequality that involves a rational expression.
Q: Can I use a graphing calculator to solve rational inequalities?
A: Yes, you can use a graphing calculator to solve rational inequalities. However, it's always a good idea to check your work by hand to make sure that you get the correct answer.
Q: How do I express the answer to a rational inequality in interval form?
A: To express the answer to a rational inequality in interval form, you need to identify the values of x that make the rational expression true or false. Then, you can use interval notation to express the solution.
Q: What is interval notation?
A: Interval notation is a way of expressing a solution set using parentheses and brackets. For example, the solution set can be expressed in interval notation as .
Q: Can I use a rational inequality to solve a rational equation?
A: Yes, you can use a rational inequality to solve a rational equation. However, you need to make sure that the rational inequality is true for all values of x that make the rational equation true.
Q: How do I know if a rational inequality is linear or nonlinear?
A: To determine if a rational inequality is linear or nonlinear, you need to look at the degree of the numerator and denominator. If the degree of the numerator and denominator are the same, then the rational inequality is linear. If the degree of the numerator and denominator are different, then the rational inequality is nonlinear.
Q: Can I use a rational inequality to solve a system of equations?
A: Yes, you can use a rational inequality to solve a system of equations. However, you need to make sure that the rational inequality is true for all values of x that make the system of equations true.
Conclusion
We hope that this article has helped you understand some of the most frequently asked questions about rational inequalities. Remember, rational inequalities can be a challenging topic, but with practice and patience, you can master them. If you have any more questions or need further clarification, please don't hesitate to ask.
Frequently Asked Questions
Q: What is the difference between a rational inequality and a rational equation?
A: A rational equation is an equation that involves a rational expression, whereas a rational inequality is an inequality that involves a rational expression.
Q: Can I use a graphing calculator to solve rational inequalities?
A: Yes, you can use a graphing calculator to solve rational inequalities. However, it's always a good idea to check your work by hand to make sure that you get the correct answer.
Q: How do I express the answer to a rational inequality in interval form?
A: To express the answer to a rational inequality in interval form, you need to identify the values of x that make the rational expression true or false. Then, you can use interval notation to express the solution.
References
- [1] "Rational Inequalities" by Math Open Reference
- [2] "Solving Rational Inequalities" by Khan Academy
Discussion
The solution to a rational inequality can be a challenging topic for many students. However, with practice and patience, you can master it. Remember, rational inequalities can be used to solve a wide range of problems, including rational equations and systems of equations.
Answer
A. is incorrect B. is correct C. is incorrect D. is not a valid option