Describe The Steps You Would Use To Solve The Following Inequality:${ \frac{x+1}{2x-3} \ \textgreater \ 2 }$

Mar 11, 2025 by ADMIN 112 views

Solving the Inequality: A Step-by-Step Guide

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Introduction

In this article, we will explore the steps involved in solving the given inequality: $\frac{x+1}{2x-3} > 2$ . This inequality involves a rational expression, and our goal is to isolate the variable $x$ and determine the values of $x$ that satisfy the inequality.

Step 1: Subtract 2 from Both Sides

The first step in solving the inequality is to subtract 2 from both sides of the equation. This will help us isolate the rational expression on one side of the inequality.

$\frac{x+1}{2x-3} - 2 > 0$

Step 2: Find a Common Denominator

To simplify the inequality, we need to find a common denominator for the two terms. The common denominator is $(2x-3)$ .

$\frac{x+1}{2x-3} - \frac{2(2x-3)}{2x-3} > 0$

Step 3: Simplify the Inequality

Now that we have a common denominator, we can simplify the inequality by combining the two terms.

$\frac{x+1-2(2x-3)}{2x-3} > 0$

Step 4: Expand and Simplify

Expanding and simplifying the numerator, we get:

$\frac{x+1-4x+6}{2x-3} > 0$

Step 5: Combine Like Terms

Combining like terms in the numerator, we get:

$\frac{-3x+7}{2x-3} > 0$

Step 6: Find the Critical Points

To solve the inequality, we need to find the critical points. The critical points are the values of $x$ that make the numerator and denominator equal to zero.

Setting the numerator equal to zero, we get:

$-3x+7 = 0$

Solving for $x$ , we get:

$x = \frac{7}{3}$

Setting the denominator equal to zero, we get:

$2x-3 = 0$

Solving for $x$ , we get:

$x = \frac{3}{2}$

Step 7: Test the Intervals

Now that we have the critical points, we need to test the intervals to determine which intervals satisfy the inequality.

The intervals are:

$(-\infty, \frac{3}{2})$
$(\frac{3}{2}, \frac{7}{3})$
$(\frac{7}{3}, \infty)$

Step 8: Test a Value from Each Interval

To determine which intervals satisfy the inequality, we need to test a value from each interval.

Let's test a value from each interval:

From the interval $(-\infty, \frac{3}{2})$ , let's test $x = 0$ .
From the interval $(\frac{3}{2}, \frac{7}{3})$ , let's test $x = 2$ .
From the interval $(\frac{7}{3}, \infty)$ , let's test $x = 4$ .

Step 9: Determine the Solution Set

After testing a value from each interval, we can determine which intervals satisfy the inequality.

The intervals that satisfy the inequality are:

$(\frac{3}{2}, \frac{7}{3})$

Conclusion

In this article, we have solved the inequality $\frac{x+1}{2x-3} > 2$ using a step-by-step approach. We have found the critical points, tested the intervals, and determined the solution set. The solution set is $(\frac{3}{2}, \frac{7}{3})$ .

Final Answer

The final answer is $\boxed{(\frac{3}{2}, \frac{7}{3})}$ .

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Introduction

In the previous article, we explored the steps involved in solving the inequality $\frac{x+1}{2x-3} > 2$ . In this article, we will address some frequently asked questions (FAQs) about solving inequalities.

Q: What is an inequality?

A: An inequality is a statement that two expressions are not equal. It can be written in the form of $a > b$ , $a < b$ , $a \geq b$ , or $a \leq b$ .

Q: What are the steps to solve an inequality?

A: The steps to solve an inequality are:

Subtract or add the same value to both sides of the inequality.
Multiply or divide both sides of the inequality by the same non-zero value.
Find the critical points by setting the numerator and denominator equal to zero.
Test the intervals to determine which intervals satisfy the inequality.
Determine the solution set based on the intervals that satisfy the inequality.

Q: What are critical points?

A: Critical points are the values of $x$ that make the numerator and denominator equal to zero. They are used to divide the number line into intervals and test each interval to determine which ones satisfy the inequality.

Q: How do I test an interval?

A: To test an interval, choose a value from the interval and substitute it into the inequality. If the inequality is true, then the interval satisfies the inequality.

Q: What is the solution set?

A: The solution set is the set of all values of $x$ that satisfy the inequality. It is determined by testing the intervals and selecting the intervals that satisfy the inequality.

Q: Can I use the same steps to solve a system of inequalities?

A: Yes, you can use the same steps to solve a system of inequalities. However, you will need to consider the intersection of the solution sets of each inequality.

Q: How do I graph an inequality?

A: To graph an inequality, you can use a number line or a coordinate plane. Mark the critical points on the number line or coordinate plane and test each interval to determine which ones satisfy the inequality.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality. However, you will need to enter the inequality in the correct format and follow the instructions on the calculator to solve it.

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

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

Not following the order of operations
Not considering the signs of the numbers
Not testing all intervals
Not considering the intersection of the solution sets of each inequality

Conclusion

In this article, we have addressed some frequently asked questions (FAQs) about solving inequalities. We have covered topics such as what an inequality is, the steps to solve an inequality, critical points, testing intervals, the solution set, and graphing inequalities. We hope this article has been helpful in answering your questions about solving inequalities.

Final Answer

The final answer is $\boxed{(\frac{3}{2}, \frac{7}{3})}$ .