Solve The Inequality:$\[ \frac{x+7}{x-4} - \frac{x-7}{x+4} \leq 0 \\]The Solution Set Is \[$\square\$\].(Type The Solution Set In Interval Notation.)

Feb 28, 2025 by ADMIN 150 views

Solving the Inequality: A Step-by-Step Guide

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Introduction

In this article, we will delve into the world of inequalities and explore a specific problem that requires us to find the solution set in interval notation. The given inequality is $\frac{x+7}{x-4} - \frac{x-7}{x+4} \leq 0$ . Our goal is to solve this inequality and express the solution set in interval notation.

Understanding the Inequality

Before we dive into the solution, let's break down the given inequality and understand its components. We have two fractions, $\frac{x+7}{x-4}$ and $\frac{x-7}{x+4}$ , which are being subtracted. The inequality states that the result of this subtraction is less than or equal to zero.

Combining the Fractions

To simplify the inequality, we can combine the two fractions by finding a common denominator. The common denominator is $(x-4)(x+4)$ , which is the product of the denominators of the two fractions.

\frac{(x+7)(x+4) - (x-7)(x-4)}{(x-4)(x+4)} \leq 0

Expanding the Numerator

Now, let's expand the numerator of the combined fraction.

\frac{x^2 + 11x + 28 - (x^2 - 11x + 28)}{(x-4)(x+4)} \leq 0

Simplifying the Numerator

After expanding the numerator, we can simplify it by combining like terms.

\frac{2x^2 + 22x - 28}{(x-4)(x+4)} \leq 0

Factoring the Numerator

Next, let's factor the numerator to see if we can simplify the inequality further.

\frac{2(x^2 + 11x - 14)}{(x-4)(x+4)} \leq 0

Factoring the Quadratic Expression

Now, let's factor the quadratic expression in the numerator.

\frac{2(x-2)(x+7)}{(x-4)(x+4)} \leq 0

Creating a Sign Table

To solve the inequality, we can create a sign table to analyze the signs of the factors in different intervals.

Interval	$x-2$	$x+7$	$x-4$	$x+4$	$\frac{2(x-2)(x+7)}{(x-4)(x+4)}$
$(-\infty, -7)$	-	-	-	-	-
$(-7, -4)$	-	-	-	+	+
$(-4, 2)$	-	+	-	+	-
$(2, 4)$	+	+	-	+	+
$(4, \infty)$	+	+	+	+	+

Analyzing the Sign Table

From the sign table, we can see that the expression $\frac{2(x-2)(x+7)}{(x-4)(x+4)}$ is negative in the interval $(-4, 2)$ .

Conclusion

Therefore, the solution set to the inequality $\frac{x+7}{x-4} - \frac{x-7}{x+4} \leq 0$ is $(-4, 2]$ .

Final Answer

The final answer is $\boxed{(-4, 2]}$ .

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Q: What is the first step in solving the inequality $\frac{x+7}{x-4} - \frac{x-7}{x+4} \leq 0$ ?

A: The first step in solving the inequality is to combine the two fractions by finding a common denominator.

Q: How do I find the common denominator for the two fractions?

A: To find the common denominator, we multiply the denominators of the two fractions, which are $(x-4)$ and $(x+4)$ .

Q: What is the next step after finding the common denominator?

A: After finding the common denominator, we can rewrite the inequality with the combined fraction and then expand the numerator.

Q: How do I expand the numerator of the combined fraction?

A: To expand the numerator, we multiply the two binomials in the numerator, $(x+7)(x+4)$ and $(x-7)(x-4)$ .

Q: What is the result of expanding the numerator?

A: The result of expanding the numerator is $x^2 + 11x + 28 - (x^2 - 11x + 28)$ .

Q: How do I simplify the numerator?

A: To simplify the numerator, we combine like terms, which results in $2x^2 + 22x - 28$ .

Q: What is the next step after simplifying the numerator?

A: After simplifying the numerator, we can factor the quadratic expression to see if we can simplify the inequality further.

Q: How do I factor the quadratic expression?

A: To factor the quadratic expression, we look for two numbers whose product is $2(-28) = -56$ and whose sum is $22$ . These numbers are $14$ and $-4$ , so we can factor the quadratic expression as $2(x^2 + 11x - 14)$ .

Q: What is the next step after factoring the quadratic expression?

A: After factoring the quadratic expression, we can rewrite the inequality with the factored numerator and then create a sign table to analyze the signs of the factors in different intervals.

Q: How do I create a sign table?

A: To create a sign table, we identify the critical points of the inequality, which are the values of $x$ that make each factor equal to zero. We then create a table with the intervals between these critical points and determine the sign of each factor in each interval.

Q: What is the final step in solving the inequality?

A: The final step in solving the inequality is to analyze the sign table and determine the intervals where the inequality is true.

Q: What is the solution set to the inequality?

A: The solution set to the inequality is $(-4, 2]$ .

Q: Why is the solution set $(-4, 2]$ ?

A: The solution set is $(-4, 2]$ because the inequality is true for all values of $x$ in this interval, except for $x = 2$ , which is not included in the solution set.

Q: What is the final answer to the inequality?

A: The final answer to the inequality is $\boxed{(-4, 2]}$ .

Q: Can you provide more examples of inequalities that can be solved using this method?

A: Yes, here are a few examples of inequalities that can be solved using this method:

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

Q: How do I know which method to use when solving an inequality?

A: When solving an inequality, you should first try to simplify the inequality by combining like terms and factoring the numerator. If this does not work, you can try creating a sign table to analyze the signs of the factors in different intervals.