Solve The Inequality:${ X - \frac{15}{x} \ \textless \ -2 }$

Solving Inequalities: A Step-by-Step Guide to Solving the Inequality $x - \frac{15}{x} < -2$

Inequalities are mathematical expressions that compare two values, and solving them is an essential skill in mathematics. In this article, we will focus on solving the inequality $x - \frac{15}{x} < -2$. We will break down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer.

Understanding the Inequality

The given inequality is $x - \frac{15}{x} < -2$. To begin solving this inequality, we need to understand the concept of inequalities and how to manipulate them. An inequality is a mathematical statement that compares two values, and it can be either greater than, less than, greater than or equal to, or less than or equal to.

In this case, we have a rational expression on the left-hand side, which is $x - \frac{15}{x}$. Our goal is to isolate the variable $x$ and determine the values of $x$ that satisfy the inequality.

Step 1: Move the Constant Term to the Right-Hand Side

To begin solving the inequality, we can start by moving the constant term to the right-hand side. This will give us a better understanding of the inequality and make it easier to manipulate.

$x - \frac{15}{x} < -2$

Subtracting $-2$ from both sides gives us:

$x - \frac{15}{x} + 2 < 0$

Simplifying the left-hand side, we get:

$x + \frac{2}{x} - \frac{15}{x} < 0$

Combining like terms, we have:

$x + \frac{-13}{x} < 0$

Step 2: Combine the Fractions

Now that we have the inequality in the form $x + \frac{-13}{x} < 0$, we can combine the fractions on the left-hand side.

$x + \frac{-13}{x} = \frac{x^2 - 13}{x}$

So, the inequality becomes:

$\frac{x^2 - 13}{x} < 0$

Step 3: Find the Critical Points

To solve the inequality, we need to find the critical points, which are the values of $x$ that make the numerator or denominator equal to zero.

Setting the numerator equal to zero, we get:

$x^2 - 13 = 0$

Solving for $x$, we have:

$x^2 = 13$

$x = \pm \sqrt{13}$

Setting the denominator equal to zero, we get:

$x = 0$

Step 4: Create a Sign Chart

Now that we have the critical points, we can create a sign chart to determine the intervals where the inequality is true.

Interval	Sign of $x^2 - 13$	Sign of $\frac{1}{x}$	Sign of $\frac{x^2 - 13}{x}$
$(-\infty, -\sqrt{13})$	-	-	+
$(-\sqrt{13}, 0)$	-	+	-
$(0, \sqrt{13})$	+	+	+
$(\sqrt{13}, \infty)$	+	-	-

Step 5: Determine the Solution

Based on the sign chart, we can determine the intervals where the inequality is true.

The inequality is true when $\frac{x^2 - 13}{x} < 0$, which occurs when $x$ is in the interval $(-\sqrt{13}, 0)$ or $(0, \sqrt{13})$.

In this article, we solved the inequality $x - \frac{15}{x} < -2$ using algebraic manipulations and mathematical concepts. We broke down the solution into manageable steps, using a sign chart to determine the intervals where the inequality is true.

The final answer is:

The solution to the inequality $x - \frac{15}{x} < -2$ is $x \in (-\sqrt{13}, 0) \cup (0, \sqrt{13})$
Solving Inequalities: A Q&A Guide to Solving the Inequality $x - \frac{15}{x} < -2$

In our previous article, we solved the inequality $x - \frac{15}{x} < -2$ using algebraic manipulations and mathematical concepts. In this article, we will provide a Q&A guide to help you better understand the solution and answer any questions you may have.

Q: What is the first step in solving the inequality $x - \frac{15}{x} < -2$?

A: The first step in solving the inequality $x - \frac{15}{x} < -2$ is to move the constant term to the right-hand side. This will give us a better understanding of the inequality and make it easier to manipulate.

Q: How do I combine the fractions on the left-hand side of the inequality?

A: To combine the fractions on the left-hand side of the inequality, we need to find a common denominator. In this case, the common denominator is $x$. So, we can rewrite the inequality as $\frac{x^2 - 13}{x} < 0$.

Q: What are the critical points of the inequality?

A: The critical points of the inequality are the values of $x$ that make the numerator or denominator equal to zero. In this case, the critical points are $x = \pm \sqrt{13}$ and $x = 0$.

Q: How do I create a sign chart to determine the intervals where the inequality is true?

A: To create a sign chart, we need to determine the sign of the numerator and denominator in each interval. We can do this by testing a value in each interval and determining the sign of the expression. The sign chart will help us determine the intervals where the inequality is true.

Q: What are the intervals where the inequality is true?

A: Based on the sign chart, we can determine the intervals where the inequality is true. In this case, the inequality is true when $x$ is in the interval $(-\sqrt{13}, 0)$ or $(0, \sqrt{13})$.

Q: What is the final answer to the inequality $x - \frac{15}{x} < -2$?

A: The final answer to the inequality $x - \frac{15}{x} < -2$ is $x \in (-\sqrt{13}, 0) \cup (0, \sqrt{13})$.

• Q: What is the difference between a rational expression and a rational inequality? A: A rational expression is an expression that contains a fraction, while a rational inequality is an inequality that contains a fraction.
• Q: How do I determine the sign of a rational expression? A: To determine the sign of a rational expression, we need to determine the sign of the numerator and denominator. We can do this by testing a value in each interval and determining the sign of the expression.
• Q: What is the purpose of a sign chart in solving rational inequalities? A: The purpose of a sign chart is to help us determine the intervals where the inequality is true. By testing a value in each interval and determining the sign of the expression, we can determine the intervals where the inequality is true.

In this article, we provided a Q&A guide to help you better understand the solution to the inequality $x - \frac{15}{x} < -2$. We answered common questions and provided additional information to help you solve rational inequalities.