Factorizing the Denominator

Oct 13, 2025 by windailys.com 95 views

Introduction

In this article, we will delve into solving the inequality $\frac{x-1}{x^2-4} < 0$ . This involves finding the intervals of $x$ where the given inequality holds true. To do this, we need to factorize the denominator and then use the concept of sign chart to determine the intervals where the inequality is satisfied.

Factorizing the Denominator

The denominator of the given expression is $x^2-4$ . We can factorize this expression as follows:

$x^2-4 = (x-2)(x+2)$

So, the given inequality can be rewritten as:

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

Sign Chart

To determine the intervals where the inequality is satisfied, we need to create a sign chart. This involves finding the critical points where the expression changes sign.

The critical points are the values of $x$ where the expression $(x-2)(x+2)$ changes sign. These points are $x=2$ and $x=-2$ .

We also need to consider the point where the numerator $x-1$ changes sign. This point is $x=1$ .

Creating the Sign Chart

Now, let's create the sign chart for the expression $\frac{x-1}{(x-2)(x+2)}$ .

Interval	$(x-2)$	$(x+2)$	$x-1$	$\frac{x-1}{(x-2)(x+2)}$
$(-\infty, -2)$	$-$	$-$	$-$	$+$
$(-2, -1)$	$-$	$-$	$+$	$-$
$(-1, 1)$	$-$	$+$	$+$	$-$
$(1, 2)$	$+$	$+$	$+$	$+$
$(2, \infty)$	$+$	$+$	$+$	$+$

Determining the Intervals

From the sign chart, we can see that the expression $\frac{x-1}{(x-2)(x+2)}$ is negative in the intervals $(-2, -1)$ and $(1, 2)$ .

Therefore, the solution to the inequality $\frac{x-1}{x^2-4} < 0$ is:

$(-2, -1) \cup (1, 2)$

Conclusion

In this article, we solved the inequality $\frac{x-1}{x^2-4} < 0$ by factorizing the denominator and creating a sign chart. We determined the intervals where the inequality is satisfied and found the solution to be $(-2, -1) \cup (1, 2)$ .

Frequently Asked Questions

Q: What is the solution to the inequality $\frac{x-1}{x^2-4} < 0$ ? A: The solution to the inequality is $(-2, -1) \cup (1, 2)$ .

Q: How do we determine the intervals where the inequality is satisfied? A: We create a sign chart by finding the critical points where the expression changes sign and then determine the intervals where the expression is negative.

References

[1] [Book Title], [Author], [Publisher], [Year].
[2] [Book Title], [Author], [Publisher], [Year].

Note: The references are fictional and for demonstration purposes only.

Introduction

In our previous article, we solved the inequality $\frac{x-1}{x^2-4} < 0$ by factorizing the denominator and creating a sign chart. In this article, we will answer some frequently asked questions related to solving inequalities.

Q&A

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to simplify the expression by combining like terms and eliminating any fractions.

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

A: To determine the critical points of an inequality, you need to find the values of the variable that make the numerator and denominator equal to zero.

Q: What is a sign chart?

A: A sign chart is a table that shows the sign of the expression in different intervals. It is used to determine the intervals where the inequality is satisfied.

Q: How do I create a sign chart?

A: To create a sign chart, you need to identify the critical points of the inequality and then determine the sign of the expression in each interval.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b < c$ or $ax + b > c$ , where $a$ , $b$ , and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c < 0$ or $ax^2 + bx + c > 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign chart to determine the intervals where the inequality is satisfied.

Q: What is the solution to the inequality $\frac{x+1}{x^2-9} > 0$ ?

A: To solve the inequality $\frac{x+1}{x^2-9} > 0$ , you need to factor the denominator and then create a sign chart. The solution to the inequality is $(-\infty, -3) \cup (-1, 3)$ .

Q: How do I determine the intervals where the inequality is satisfied?

A: To determine the intervals where the inequality is satisfied, you need to use the sign chart to identify the intervals where the expression is positive or negative.

Q: What is the difference between a rational inequality and a polynomial inequality?

A: A rational inequality is an inequality that can be written in the form $\frac{f(x)}{g(x)} < 0$ or $\frac{f(x)}{g(x)} > 0$ , where $f(x)$ and $g(x)$ are polynomials. A polynomial inequality is an inequality that can be written in the form $f(x) < 0$ or $f(x) > 0$ , where $f(x)$ is a polynomial.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to factor the numerator and denominator and then use the sign chart to determine the intervals where the inequality is satisfied.

Conclusion

In this article, we answered some frequently asked questions related to solving inequalities. We covered topics such as simplifying expressions, determining critical points, creating sign charts, and solving linear and quadratic inequalities.

Frequently Asked Questions

Q: What is the first step in solving an inequality? A: The first step in solving an inequality is to simplify the expression by combining like terms and eliminating any fractions.

Q: How do I determine the critical points of an inequality? A: To determine the critical points of an inequality, you need to find the values of the variable that make the numerator and denominator equal to zero.

References

[1] [Book Title], [Author], [Publisher], [Year].
[2] [Book Title], [Author], [Publisher], [Year].

Note: The references are fictional and for demonstration purposes only.