Solve The Inequality:1. \[$\frac{x+2}{2x} \leq 0\$\]2. \[$\frac{x-2}{2x} \leq 0\$\]

Feb 28, 2025 by ADMIN 84 views

**Solving Inequalities: A Step-by-Step Guide**

In this article, we will delve into the world of inequalities and explore two specific cases: $\frac{x+2}{2x} \leq 0$ and $\frac{x-2}{2x} \leq 0$ . We will break down each inequality, identify the critical points, and determine the intervals where the inequality holds true.

Understanding Inequalities

An inequality is a statement that compares two expressions using a mathematical operator, such as less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). Inequalities can be linear, quadratic, or more complex, and they can be used to model a wide range of real-world problems.

Case 1: $\frac{x+2}{2x} \leq 0$

To solve this inequality, we need to find the values of $x$ that make the expression $\frac{x+2}{2x}$ less than or equal to zero.

Step 1: Find the Critical Points

The critical points are the values of $x$ that make the numerator or denominator equal to zero. In this case, the critical points are $x = -2$ (numerator) and $x = 0$ (denominator).

Step 2: Create a Sign Chart

To determine the intervals where the inequality holds true, we need to create a sign chart. We will examine the signs of the factors in each interval.

Interval	$x + 2$	$2x$	$\frac{x+2}{2x}$
$(-\infty, -2)$	-	-	+
$(-2, 0)$	+	-	-
$(0, \infty)$	+	+	+

Step 3: Determine the Solution Intervals

Based on the sign chart, we can see that the inequality holds true in the interval $(-2, 0)$ .

Case 2: $\frac{x-2}{2x} \leq 0$

To solve this inequality, we need to find the values of $x$ that make the expression $\frac{x-2}{2x}$ less than or equal to zero.

Step 4: Find the Critical Points

The critical points are the values of $x$ that make the numerator or denominator equal to zero. In this case, the critical points are $x = 2$ (numerator) and $x = 0$ (denominator).

Step 5: Create a Sign Chart

To determine the intervals where the inequality holds true, we need to create a sign chart. We will examine the signs of the factors in each interval.

Interval	$x - 2$	$2x$	$\frac{x-2}{2x}$
$(-\infty, 0)$	-	-	+
$(0, 2)$	-	+	-
$(2, \infty)$	+	+	+

Step 6: Determine the Solution Intervals

Based on the sign chart, we can see that the inequality holds true in the interval $(0, 2)$ .

Conclusion

In this article, we have solved two inequalities: $\frac{x+2}{2x} \leq 0$ and $\frac{x-2}{2x} \leq 0$ . We have identified the critical points, created sign charts, and determined the solution intervals for each inequality. By following these steps, we can solve a wide range of inequalities and gain a deeper understanding of mathematical concepts.

Tips and Tricks

When solving inequalities, it's essential to find the critical points and create a sign chart to determine the solution intervals.
Be careful when dealing with fractions, as they can be tricky to work with.
Practice, practice, practice! Solving inequalities takes time and practice to become proficient.

Real-World Applications

Inequalities have numerous real-world applications, including:

Modeling population growth and decline
Analyzing financial data and making investment decisions
Solving optimization problems in fields like engineering and economics
Understanding and predicting natural phenomena, such as weather patterns and climate change

In our previous article, we explored the world of inequalities and solved two specific cases: $\frac{x+2}{2x} \leq 0$ and $\frac{x-2}{2x} \leq 0$ . In this article, we will answer some of the most frequently asked questions about solving inequalities.

Q: What is an inequality?

A: An inequality is a statement that compares two expressions using a mathematical operator, such as less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).

Q: How do I solve an inequality?

A: To solve an inequality, you need to find the values of the variable that make the expression true. This involves identifying the critical points, creating a sign chart, and determining the solution intervals.

Q: What are critical points?

A: Critical points are the values of the variable that make the numerator or denominator equal to zero. These points are crucial in determining the solution intervals.

Q: How do I create a sign chart?

A: A sign chart is a table that shows the signs of the factors in each interval. To create a sign chart, you need to examine the signs of the factors in each interval and determine the overall sign of the expression.

Q: What are the different types of inequalities?

A: There are several types of inequalities, including:

Linear inequalities: These are inequalities that involve a linear expression, such as $x + 2 \leq 0$ .
Quadratic inequalities: These are inequalities that involve a quadratic expression, such as $x^2 + 2x + 1 \leq 0$ .
Rational inequalities: These are inequalities that involve a rational expression, such as $\frac{x+2}{2x} \leq 0$ .

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to find the values of the variable that make the expression true. This involves identifying the critical points and determining the solution intervals.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to find the values of the variable that make the expression true. This involves identifying the critical points, creating a sign chart, and determining the solution intervals.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to find the values of the variable that make the expression true. This involves identifying the critical points, creating a sign chart, and determining the solution intervals.

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

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

Not identifying the critical points
Not creating a sign chart
Not determining the solution intervals
Not checking the solution intervals for extraneous solutions

Q: How do I check my solution intervals for extraneous solutions?

A: To check your solution intervals for extraneous solutions, you need to plug in a value from each interval into the original inequality and check if it is true.

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

A: Inequalities have numerous real-world applications, including:

Modeling population growth and decline
Analyzing financial data and making investment decisions
Solving optimization problems in fields like engineering and economics
Understanding and predicting natural phenomena, such as weather patterns and climate change

By mastering the art of solving inequalities, you can unlock a wide range of mathematical and real-world applications, and gain a deeper understanding of the world around you.