Solve The Rational Inequality And Graph The Solution Set On A Real Number Line. Express The Solution Set In Interval Notation.${ \frac{(x+8)(x-10)}{x+6} \leq 0 }$1. Move All Terms To The Left Side Of The Inequality And Define The Left Side

Introduction

Rational inequalities are a type of mathematical expression that involves a rational function, which is a ratio of two polynomials. Solving rational inequalities requires a combination of algebraic and graphical techniques to find the solution set. In this article, we will focus on solving the rational inequality $\frac{(x+8)(x-10)}{x+6} \leq 0$ and graphing the solution set on a real number line.

Step 1: Move All Terms to the Left Side of the Inequality

The first step in solving a rational inequality is to move all terms to the left side of the inequality. This will allow us to define the left side of the inequality and analyze its behavior.

$\frac{(x+8)(x-10)}{x+6} \leq 0$

To move all terms to the left side, we can multiply both sides of the inequality by $x+6$, which is the denominator of the rational function. However, we must be careful when multiplying both sides of an inequality by a variable expression, as this can change the direction of the inequality.

$(x+8)(x-10) \leq 0(x+6)$

Since $0(x+6) = 0$, we can simplify the inequality to:

$(x+8)(x-10) \leq 0$

Step 2: Define the Left Side of the Inequality

The left side of the inequality is a quadratic expression, which can be factored as:

$(x+8)(x-10) = x^2 - 2x - 80$

This quadratic expression has two roots, which are the values of $x$ that make the expression equal to zero. The roots of the quadratic expression are $x = -8$ and $x = 10$.

Step 3: Find the Critical Points

The critical points of the rational function are the values of $x$ that make the numerator or denominator equal to zero. In this case, the critical points are $x = -8$, $x = 10$, and $x = -6$.

Step 4: Test the Intervals

To find the solution set, we need to test the intervals between the critical points. We can do this by choosing a test point in each interval and evaluating the rational function at that point.

Interval 1: $(-\infty, -8)$

Let's choose a test point in this interval, such as $x = -10$. Evaluating the rational function at this point, we get:

$\frac{(-10+8)(-10-10)}{-10+6} = \frac{(-2)(-20)}{-4} = 10$

Since the rational function is positive in this interval, we can conclude that the solution set does not include this interval.

Interval 2: $(-8, -6)$

Let's choose a test point in this interval, such as $x = -7$. Evaluating the rational function at this point, we get:

$\frac{(-7+8)(-7-10)}{-7+6} = \frac{(1)(-17)}{-1} = 17$

Since the rational function is positive in this interval, we can conclude that the solution set does not include this interval.

Interval 3: $(-6, 10)$

Let's choose a test point in this interval, such as $x = 0$. Evaluating the rational function at this point, we get:

$\frac{(0+8)(0-10)}{0+6} = \frac{(8)(-10)}{6} = -\frac{80}{6} = -\frac{40}{3}$

Since the rational function is negative in this interval, we can conclude that the solution set includes this interval.

Interval 4: $(10, \infty)$

Let's choose a test point in this interval, such as $x = 12$. Evaluating the rational function at this point, we get:

$\frac{(12+8)(12-10)}{12+6} = \frac{(20)(2)}{18} = \frac{40}{18} = \frac{20}{9}$

Since the rational function is positive in this interval, we can conclude that the solution set does not include this interval.

Conclusion

In conclusion, the solution set to the rational inequality $\frac{(x+8)(x-10)}{x+6} \leq 0$ is the interval $(-\infty, -8) \cup (-6, 10)$. This can be expressed in interval notation as $(-\infty, -8) \cup (-6, 10)$.

Graphing the Solution Set

To graph the solution set, we can use a number line and mark the critical points $x = -8$, $x = -6$, and $x = 10$. We can then test the intervals between these points and mark the solution set accordingly.

The final graph of the solution set is a union of two intervals, $(-\infty, -8)$ and $(-6, 10)$.

Final Answer

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Q&A: Frequently Asked Questions

Q: What is a rational inequality?

A: A rational inequality is a type of mathematical expression that involves a rational function, which is a ratio of two polynomials.

Q: How do I solve a rational inequality?

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

1. Move all terms to the left side of the inequality.
2. Define the left side of the inequality.
3. Find the critical points of the rational function.
4. Test the intervals between the critical points.
5. Express the solution set in interval notation.

Q: What are critical points?

A: Critical points are the values of $x$ that make the numerator or denominator of the rational function equal to zero.

Q: How do I test the intervals?

A: To test the intervals, you need to choose a test point in each interval and evaluate the rational function at that point.

Q: What is the solution set?

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

Q: How do I express the solution set in interval notation?

A: To express the solution set in interval notation, you need to use the following notation:

• $(-\infty, a)$: all values of $x$ less than $a$
• $(a, b)$: all values of $x$ between $a$ and $b$
• $[a, b]$ : all values of $x$ between $a$ and $b$, including $a$ and $b$
• $(a, \infty)$: all values of $x$ greater than $a$

Q: What is the final answer?

A: The final answer is the solution set expressed in interval notation.

Example: Solving a Rational Inequality

Let's solve the rational inequality $\frac{(x+8)(x-10)}{x+6} \leq 0$.

Step 1: Move all terms to the left side of the inequality

$\frac{(x+8)(x-10)}{x+6} \leq 0$

Step 2: Define the left side of the inequality

$(x+8)(x-10) \leq 0$

Step 3: Find the critical points

$x = -8$, $x = 10$, and $x = -6$

Step 4: Test the intervals

Interval 1: $(-\infty, -8)$ Interval 2: $(-8, -6)$ Interval 3: $(-6, 10)$ Interval 4: $(10, \infty)$

Step 5: Express the solution set in interval notation

The solution set is $(-\infty, -8) \cup (-6, 10)$.

Conclusion

In conclusion, solving rational inequalities requires a combination of algebraic and graphical techniques. By following the steps outlined in this article, you can solve rational inequalities and express the solution set in interval notation.

Final Answer

The final answer is $\boxed{(-\infty, -8) \cup (-6, 10)}$.