What Is The Solution Of 2 X − 1 \textgreater 7 X + 1 \frac{2}{x-1}\ \textgreater \ \frac{7}{x+1} X − 1 2 ​ \textgreater X + 1 7 ​ ?A. { -1 \ \textless \ X \ \textless \ 1$}$ Or { X \ \textgreater \ \frac{9}{5}$}$B. { -1 \ \textless \ X \ \textless \ 1$}$ Or

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Introduction

In this article, we will explore the solution to the inequality $\frac{2}{x-1}\ \textgreater \ \frac{7}{x+1}$. This type of inequality is known as a rational inequality, and it involves fractions with variables in the numerator and denominator. To solve this inequality, we will use various algebraic techniques, including cross-multiplication and factoring.

Understanding the Inequality

The given inequality is $\frac{2}{x-1}\ \textgreater \ \frac{7}{x+1}$. This means that the fraction $\frac{2}{x-1}$ is greater than the fraction $\frac{7}{x+1}$. To begin solving this inequality, we need to get rid of the fractions by cross-multiplying.

Cross-Multiplication

To eliminate the fractions, we can multiply both sides of the inequality by the denominators of the fractions. In this case, we will multiply both sides by $(x-1)(x+1)$.

\frac{2}{x-1} > \frac{7}{x+1}

Cross-multiplying gives us:

2(x+1) > 7(x-1)

Expanding and Simplifying

Now, we can expand and simplify the inequality by distributing the numbers to the terms inside the parentheses.

2x + 2 > 7x - 7

Subtracting $2x$ from both sides gives us:

2 > 5x - 7

Adding $7$ to both sides gives us:

9 > 5x

Solving for x

Now, we can solve for $x$ by dividing both sides of the inequality by $5$.

\frac{9}{5} > x

Finding the Solution Set

The solution set to the inequality $\frac{2}{x-1}\ \textgreater \ \frac{7}{x+1}$ is the set of all values of $x$ that satisfy the inequality. In this case, the solution set is $x \in \left(-\infty, -1\right) \cup \left(\frac{9}{5}, \infty\right)$.

Conclusion

In conclusion, the solution to the inequality $\frac{2}{x-1}\ \textgreater \ \frac{7}{x+1}$ is $x \in \left(-\infty, -1\right) \cup \left(\frac{9}{5}, \infty\right)$. This means that the values of $x$ that satisfy the inequality are all real numbers less than $-1$ and all real numbers greater than $\frac{9}{5}$.

Final Answer

The final answer is $\boxed{\left(-\infty, -1\right) \cup \left(\frac{9}{5}, \infty\right)}$.

Discussion

This type of inequality is known as a rational inequality, and it involves fractions with variables in the numerator and denominator. To solve this inequality, we used various algebraic techniques, including cross-multiplication and factoring. The solution set to the inequality is the set of all values of $x$ that satisfy the inequality.

Related Topics

• Rational inequalities
• Cross-multiplication
• Factoring
• Algebraic techniques

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Precalculus" by Michael Sullivan

Keywords

• Rational inequality
• Cross-multiplication
• Factoring
• Algebraic techniques
• Solution set
• Inequality
• Rational expression
• Algebra
• Trigonometry
• Precalculus
• College algebra
  

Introduction

In our previous article, we explored the solution to the rational inequality $\frac{2}{x-1}\ \textgreater \ \frac{7}{x+1}$. In this article, we will answer some frequently asked questions about rational inequalities and provide additional examples to help you understand the concept.

Q&A

Q: What is a rational inequality?

A: A rational inequality is an inequality that involves fractions with variables in the numerator and denominator. It is a type of inequality that requires special techniques to solve.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to follow these steps:

1. Get rid of the fractions by cross-multiplying.
2. Expand and simplify the inequality.
3. Solve for the variable.
4. Find the solution set.

Q: What is cross-multiplication?

A: Cross-multiplication is a technique used to eliminate fractions in an inequality. It involves multiplying both sides of the inequality by the denominators of the fractions.

Q: How do I know when to use cross-multiplication?

A: You should use cross-multiplication when you have an inequality that involves fractions with variables in the numerator and denominator.

Q: What is the solution set?

A: The solution set is the set of all values of the variable that satisfy the inequality.

Q: How do I find the solution set?

A: To find the solution set, you need to solve for the variable and then determine the values of the variable that satisfy the inequality.

Q: Can I use the same techniques to solve rational equations?

A: Yes, you can use the same techniques to solve rational equations. However, rational equations are equalities, not inequalities, so you will need to use different techniques to solve them.

Q: What are some common mistakes to avoid when solving rational inequalities?

A: Some common mistakes to avoid when solving rational inequalities include:

• Not getting rid of the fractions by cross-multiplying.
• Not expanding and simplifying the inequality.
• Not solving for the variable.
• Not finding the solution set.

Examples

Example 1: $\frac{3}{x+2}\ \textgreater \ \frac{2}{x-1}$

To solve this inequality, we need to follow the steps outlined above.

\frac{3}{x+2} > \frac{2}{x-1}

Cross-multiplying gives us:

3(x-1) > 2(x+2)

Expanding and simplifying the inequality gives us:

3x - 3 > 2x + 4

Solving for x gives us:

x > \frac{7}{1}

The solution set is $x \in \left(\infty, \infty\right)$.

Example 2: $\frac{4}{x-3}\ \textless \ \frac{3}{x+1}$

To solve this inequality, we need to follow the steps outlined above.

\frac{4}{x-3} < \frac{3}{x+1}

Cross-multiplying gives us:

4(x+1) < 3(x-3)

Expanding and simplifying the inequality gives us:

4x + 4 < 3x - 9

Solving for x gives us:

x < -\frac{13}{1}

The solution set is $x \in \left(-\infty, -13\right)$.

Conclusion

In conclusion, rational inequalities are a type of inequality that involves fractions with variables in the numerator and denominator. To solve a rational inequality, you need to follow the steps outlined above, including cross-multiplication, expanding and simplifying, solving for the variable, and finding the solution set. By following these steps and using the examples provided, you can become proficient in solving rational inequalities.

Final Answer

The final answer is $\boxed{\left(-\infty, -13\right)}$.

Discussion

Rational inequalities are an important topic in algebra and are used to model real-world problems. By understanding how to solve rational inequalities, you can apply this knowledge to a variety of fields, including science, engineering, and economics.

Related Topics

• Rational equations
• Cross-multiplication
• Factoring
• Algebraic techniques
• Solution set
• Inequality
• Rational expression
• Algebra
• Trigonometry
• Precalculus
• College algebra

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Precalculus" by Michael Sullivan

Keywords

• Rational inequality
• Cross-multiplication
• Factoring
• Algebraic techniques
• Solution set
• Inequality
• Rational expression
• Algebra
• Trigonometry
• Precalculus
• College algebra