What Values Make The Inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$ True?

What values make the inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$ true?

In mathematics, inequalities are used to describe relationships between different values or expressions. The given inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$ is a rational inequality, where the expression on the left-hand side is a fraction. To determine the values of $x$ that make this inequality true, we need to analyze the sign of the expression and find the intervals where it is negative.

The given inequality is $\frac{x+\theta}{\theta-x}\ \textless \ 0$. This means that the expression $\frac{x+\theta}{\theta-x}$ is negative. To simplify the analysis, we can multiply both sides of the inequality by $-1$, which flips the direction of the inequality sign. This gives us $\frac{x+\theta}{\theta-x}\ \textgreater \ 0$.

To find the values of $x$ that make the inequality true, we need to find the critical points of the expression $\frac{x+\theta}{\theta-x}$. The critical points occur when the numerator or denominator of the expression is equal to zero. In this case, the critical points are $x = -\theta$ and $x = \theta$.

To determine the sign of the expression $\frac{x+\theta}{\theta-x}$, we can use a sign chart. We will analyze the sign of the expression in the intervals defined by the critical points.

• Interval 1: $x < -\theta$
• Interval 2: $-\theta < x < \theta$
• Interval 3: $x > \theta$

Based on the sign chart, we can see that the expression $\frac{x+\theta}{\theta-x}$ is negative in the interval $-\theta < x < \theta$. Therefore, the values of $x$ that make the inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$ true are $x \in (-\theta, \theta)$.

Suppose we want to find the values of $x$ that make the inequality $\frac{x+2}{2-x}\ \textless \ 0$ true. We can use the same analysis as above to find the critical points and sign chart.

• Critical points: $x = -2$ and $x = 2$
• Sign chart:

Based on the sign chart, we can see that the expression $\frac{x+2}{2-x}$ is negative in the interval $-2 < x < 2$. Therefore, the values of $x$ that make the inequality $\frac{x+2}{2-x}\ \textless \ 0$ true are $x \in (-2, 2)$.

The values of $x$ that make the inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$ true are $x \in (-\theta, \theta)$.
Frequently Asked Questions (FAQs) about the Inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$

A: The inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$ is used to determine the values of $x$ that make the expression $\frac{x+\theta}{\theta-x}$ negative. This is useful in various mathematical and real-world applications, such as solving systems of equations, finding the maximum or minimum of a function, and modeling real-world phenomena.

A: To find the critical points, set the numerator or denominator of the expression equal to zero and solve for $x$. In this case, the critical points are $x = -\theta$ and $x = \theta$.

A: The sign chart is a useful tool in analyzing the inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$. It helps us determine the sign of the expression in different intervals defined by the critical points. By analyzing the sign chart, we can determine the values of $x$ that make the inequality true.

A: Yes, the analysis used for the inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$ can be applied to other rational inequalities. The key steps are to find the critical points, create a sign chart, and analyze the sign of the expression in different intervals.

A: The inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$ can be applied to various real-world problems, such as:

• Modeling population growth or decline
• Analyzing the behavior of a system with multiple variables
• Finding the maximum or minimum of a function
• Solving systems of equations

A: Some common mistakes to avoid when analyzing the inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$ include:

• Failing to find the critical points
• Not creating a sign chart
• Misinterpreting the sign chart
• Not considering the domain of the expression

A: Yes, technology can be a useful tool in analyzing the inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$. Graphing calculators and computer software can help you visualize the expression and determine the values of $x$ that make the inequality true.

The inequality $\frac{x+\theta}{\theta-x}\ \textless \ 0$ is a useful tool in mathematics and real-world applications. By understanding the critical points, creating a sign chart, and analyzing the sign of the expression, we can determine the values of $x$ that make the inequality true.