What Is The Solution Of $\frac{x^2+x-6}{x-7} \leq 0$?A. $x \leq -3$ Or $2 \leq X \ \textless \ 7$B. $-3 \leq X \leq 2$ Or $x \ \textgreater \ 7$C. $-3 \ \textless \ X \ \textless \ 2$ Or

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Introduction

In this article, we will explore the solution to the inequality $\frac{x^2+x-6}{x-7} \leq 0$. This involves factoring the numerator, finding the critical points, and analyzing the sign of the expression in different intervals. We will also discuss the importance of understanding the behavior of rational expressions and how to apply this knowledge to solve inequalities.

Factoring the Numerator

To begin, we need to factor the numerator of the expression, which is $x^2+x-6$. We can use the quadratic formula or factor by grouping to find the factors. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation. In this case, $a=1$, $b=1$, and $c=-6$. Plugging these values into the quadratic formula, we get:

$x = \frac{-1 \pm \sqrt{1^2-4(1)(-6)}}{2(1)}$ $x = \frac{-1 \pm \sqrt{1+24}}{2}$ $x = \frac{-1 \pm \sqrt{25}}{2}$ $x = \frac{-1 \pm 5}{2}$

This gives us two possible values for $x$: $x = \frac{-1 + 5}{2} = 2$ and $x = \frac{-1 - 5}{2} = -3$. Therefore, we can factor the numerator as $(x-2)(x+3)$.

Finding the Critical Points

The critical points of the expression are the values of $x$ that make the numerator or denominator equal to zero. In this case, the critical points are $x=2$ and $x=-3$ from the numerator, and $x=7$ from the denominator. We also need to consider the point where the numerator and denominator are equal to zero, which is $x=2$.

Analyzing the Sign of the Expression

To analyze the sign of the expression, we can use a sign chart or a number line. We will consider the intervals between the critical points and determine the sign of the expression in each interval.

Interval	Sign of $(x-2)$	Sign of $(x+3)$	Sign of $(x-7)$	Sign of the Expression
$(-\infty, -3)$	-	-	-	+
$(-3, 2)$	-	+	-	-
$(2, 7)$	+	+	-	+
$(7, \infty)$	+	+	+	+

Solving the Inequality

From the sign chart, we can see that the expression is negative in the interval $(-3, 2)$. Therefore, the solution to the inequality $\frac{x^2+x-6}{x-7} \leq 0$ is $x \in (-3, 2) \cup (7, \infty)$.

Conclusion

In conclusion, we have solved the inequality $\frac{x^2+x-6}{x-7} \leq 0$ by factoring the numerator, finding the critical points, and analyzing the sign of the expression in different intervals. We have also discussed the importance of understanding the behavior of rational expressions and how to apply this knowledge to solve inequalities.

Final Answer

The final answer is $x \in (-3, 2) \cup (7, \infty)$.

Discussion

This problem is a classic example of how to solve rational inequalities. The key steps involved are factoring the numerator, finding the critical points, and analyzing the sign of the expression in different intervals. By following these steps, we can determine the solution to the inequality and understand the behavior of the rational expression.

Related Problems

• Solving rational inequalities with quadratic numerators
• Finding the critical points of rational expressions
• Analyzing the sign of rational expressions in different intervals

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Rational Expressions and Equations" by Math Open Reference
  

Introduction

Solving rational inequalities can be a challenging task, but with the right approach and techniques, it can be made easier. In this article, we will address some of the most frequently asked questions on solving rational inequalities, providing step-by-step explanations and examples to help you understand the concepts better.

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. Rational inequalities can be written in the form $\frac{f(x)}{g(x)} \leq 0$, where $f(x)$ and $g(x)$ are polynomials.

Q: How do I solve a rational inequality?

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

1. Factor the numerator and denominator of the rational expression.
2. Find the critical points of the rational expression, which are the values of $x$ that make the numerator or denominator equal to zero.
3. Analyze the sign of the rational expression in different intervals using a sign chart or a number line.
4. Determine the solution to the inequality by identifying the intervals where the rational expression is less than or equal to zero.

Q: What are the critical points of a rational expression?

A: The critical points of a rational expression are the values of $x$ that make the numerator or denominator equal to zero. These points divide the number line into intervals where the rational expression has a constant sign.

Q: How do I use a sign chart to analyze the sign of a rational expression?

A: A sign chart is a table that shows the sign of the rational expression in different intervals. To create a sign chart, you need to:

1. List the critical points of the rational expression.
2. Determine the sign of the rational expression in each interval by testing a value from each interval.
3. Fill in the sign chart with the signs of the rational expression in each interval.

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. Rational equations can be solved using algebraic methods, while rational inequalities require the use of sign charts and number lines.

Q: Can I use a calculator to solve rational inequalities?

A: While calculators can be useful for solving rational inequalities, they are not always necessary. In fact, using a calculator can sometimes lead to errors or oversimplifications. It's often better to use a sign chart or a number line to analyze the sign of the rational expression and determine the solution to the inequality.

Q: How do I determine the solution to a rational inequality?

A: To determine the solution to a rational inequality, you need to identify the intervals where the rational expression is less than or equal to zero. This can be done by analyzing the sign chart or number line and selecting the intervals where the rational expression has a negative or zero sign.

Q: Can I use the same techniques to solve rational inequalities with quadratic numerators?

A: Yes, the techniques used to solve rational inequalities with linear numerators can be applied to rational inequalities with quadratic numerators. However, you may need to use more advanced algebraic techniques, such as factoring or the quadratic formula, to solve the quadratic numerator.

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

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

• Failing to factor the numerator and denominator correctly
• Ignoring critical points or intervals
• Misinterpreting the sign chart or number line
• Failing to check for extraneous solutions

Conclusion

Solving rational inequalities requires a combination of algebraic techniques, sign charts, and number lines. By following the steps outlined in this article and avoiding common mistakes, you can successfully solve rational inequalities and gain a deeper understanding of the behavior of rational expressions.

Final Tips

• Practice, practice, practice: The more you practice solving rational inequalities, the more comfortable you will become with the techniques and the more confident you will be in your ability to solve them.
• Use a sign chart or number line: These tools can help you analyze the sign of the rational expression and determine the solution to the inequality.
• Check for extraneous solutions: Make sure to check for extraneous solutions, especially when solving rational inequalities with quadratic numerators.
• Review and revise: Review the concepts and techniques covered in this article and revise as needed to ensure you have a solid understanding of solving rational inequalities.