Which Shows All The Critical Points For The Inequality $\frac{x^2-4}{x^2-5x+6}\ \textless \ 0$?A. $x = -2$ And $x = 2$B. $x = 2$ And $x = 3$C. $x = -3$, $x = -2$, And $x = 2$D.

Mar 1, 2025 by ADMIN 178 views

**Solving Inequalities: A Step-by-Step Guide to Critical Points**

Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more expressions. Solving inequalities involves finding the values of the variable that satisfy the given inequality. One of the critical steps in solving inequalities is identifying the critical points, which are the values of the variable that make the inequality true or false. In this article, we will focus on solving the inequality $\frac{x^2-4}{x^2-5x+6}\ \textless \ 0$ and identify the critical points.

Understanding the Inequality

The given inequality is $\frac{x^2-4}{x^2-5x+6}\ \textless \ 0$ . To solve this inequality, we need to find the values of $x$ that make the expression $\frac{x^2-4}{x^2-5x+6}$ less than zero. This involves finding the critical points of the inequality, which are the values of $x$ that make the numerator or denominator equal to zero.

Finding Critical Points

To find the critical points, we need to set the numerator and denominator equal to zero and solve for $x$ . The numerator is $x^2-4$ , and the denominator is $x^2-5x+6$ . Setting the numerator equal to zero, we get:

$x^2-4 = 0$

Solving for $x$ , we get:

$x^2 = 4$

$x = \pm 2$

Setting the denominator equal to zero, we get:

$x^2-5x+6 = 0$

Solving for $x$ , we get:

$(x-2)(x-3) = 0$

$x = 2$ or $x = 3$

Analyzing the Critical Points

Now that we have found the critical points, we need to analyze them to determine which ones make the inequality true or false. To do this, we can use a sign chart or a number line to test the intervals between the critical points.

Sign Chart

Here is a sign chart for the inequality:

Interval	$x^2-4$	$x^2-5x+6$	$\frac{x^2-4}{x^2-5x+6}$
$(-\infty, -2)$	$+$	$+$	$+$
$(-2, 2)$	$+$	$-$	$-$
$(2, 3)$	$+$	$+$	$+$
$(3, \infty)$	$+$	$+$	$+$

Number Line

Here is a number line for the inequality:

$x = -2$ : The inequality is true.
$x = 2$ : The inequality is false.
$x = 3$ : The inequality is true.

Conclusion

Based on the sign chart and number line, we can conclude that the critical points of the inequality are $x = -2$ and $x = 3$ . Therefore, the correct answer is:

A. $x = -2$ and $x = 3$

Final Answer

The final answer is $\boxed{A}$ .

Additional Tips and Tricks

When solving inequalities, it's essential to identify the critical points and analyze them to determine which ones make the inequality true or false.
A sign chart or number line can be used to test the intervals between the critical points.
The critical points are the values of the variable that make the inequality true or false.
The inequality can be solved by finding the values of $x$ that make the expression $\frac{x^2-4}{x^2-5x+6}$ less than zero.

Common Mistakes to Avoid

Failing to identify the critical points.
Not analyzing the critical points to determine which ones make the inequality true or false.
Not using a sign chart or number line to test the intervals between the critical points.
Not considering the signs of the numerator and denominator when solving the inequality.

Real-World Applications

Inequalities are used in various real-world applications, such as finance, economics, and engineering.
Solving inequalities can help us make informed decisions and predictions.
Inequalities can be used to model real-world problems, such as population growth, resource allocation, and financial planning.

Conclusion

In conclusion, solving inequalities involves finding the critical points and analyzing them to determine which ones make the inequality true or false. By using a sign chart or number line, we can test the intervals between the critical points and determine the solution to the inequality. The critical points are the values of the variable that make the inequality true or false, and they are essential in solving inequalities.

Introduction

In our previous article, we discussed how to solve the inequality $\frac{x^2-4}{x^2-5x+6}\ \textless \ 0$ and identified the critical points. In this article, we will provide a Q&A guide to help you better understand the concept of solving inequalities and how to apply it to real-world problems.

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two or more expressions using a relation such as greater than, less than, greater than or equal to, or less than or equal to.

Q: What is a critical point?

A: A critical point is a value of the variable that makes the inequality true or false. Critical points are essential in solving inequalities, as they help us determine the solution to the inequality.

Q: How do I find the critical points of an inequality?

A: To find the critical points, you need to set the numerator and denominator equal to zero and solve for $x$ . This will give you the values of $x$ that make the inequality true or false.

Q: What is a sign chart?

A: A sign chart is a table that shows the signs of the expressions in the numerator and denominator of the inequality. It helps us determine the solution to the inequality by testing the intervals between the critical points.

Q: How do I use a sign chart to solve an inequality?

A: To use a sign chart, you need to identify the critical points and determine the signs of the expressions in the numerator and denominator. Then, you can use the sign chart to test the intervals between the critical points and determine the solution to the inequality.

Q: What is a number line?

A: A number line is a visual representation of the real numbers, with the critical points marked on it. It helps us determine the solution to the inequality by testing the intervals between the critical points.

Q: How do I use a number line to solve an inequality?

A: To use a number line, you need to identify the critical points and mark them on the number line. Then, you can use the number line to test the intervals between the critical points and determine the solution to the inequality.

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

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

Failing to identify the critical points.
Not analyzing the critical points to determine which ones make the inequality true or false.
Not using a sign chart or number line to test the intervals between the critical points.
Not considering the signs of the numerator and denominator when solving the inequality.

Q: How do I apply the concept of solving inequalities to real-world problems?

A: Solving inequalities can be applied to various real-world problems, such as finance, economics, and engineering. For example, you can use inequalities to model population growth, resource allocation, and financial planning.

Q: What are some real-world applications of solving inequalities?

A: Some real-world applications of solving inequalities include:

Finance: Inequalities can be used to model financial transactions, such as investments and loans.
Economics: Inequalities can be used to model economic systems, such as supply and demand.
Engineering: Inequalities can be used to model physical systems, such as motion and energy.