Solving (x+4)/(2x-1) < 0 A Step-by-Step Guide

Navigating the world of inequalities can be a daunting task, especially when dealing with rational expressions. In this article, we will embark on a journey to dissect the inequality (x+4)/(2x-1) < 0, providing a step-by-step solution and shedding light on the underlying concepts. Our primary goal is to not just provide the answer, but also to ensure a thorough understanding of the process involved. This will empower you to tackle similar problems with confidence and clarity.

Understanding the Problem: A Foundation for Success

Before we delve into the solution, let's take a moment to understand the problem at hand. The inequality (x+4)/(2x-1) < 0 presents a rational expression, where we have a fraction with expressions involving the variable 'x' in both the numerator and the denominator. Our objective is to find the range of values for 'x' that make this expression less than zero. In simpler terms, we are seeking the values of 'x' that make the fraction negative.

To effectively solve this inequality, we must consider two key factors: the numerator (x+4) and the denominator (2x-1). The sign of the entire expression hinges on the signs of these individual components. For the fraction to be negative, either the numerator must be negative and the denominator positive, or vice versa. This fundamental principle will guide our solution process.

It's also crucial to acknowledge the restriction imposed by the denominator. Since division by zero is undefined, we must exclude any value of 'x' that makes the denominator (2x-1) equal to zero. This critical point will further refine our solution set.

Step-by-Step Solution: A Clear and Concise Approach

Now, let's embark on the step-by-step solution to the inequality (x+4)/(2x-1) < 0. We will break down the process into manageable steps, ensuring clarity and comprehension at each stage.

Step 1: Finding the Critical Points

The critical points are the values of 'x' that make either the numerator or the denominator equal to zero. These points are crucial because they divide the number line into intervals where the expression's sign remains constant. To find these points, we set both the numerator and the denominator to zero and solve for 'x'.

• Numerator:
  • x + 4 = 0
  • x = -4
• Denominator:
  • 2x - 1 = 0
  • 2x = 1
  • x = 1/2

Thus, our critical points are x = -4 and x = 1/2. These points will serve as the boundaries of our intervals.

Step 2: Creating the Sign Table

The sign table is a powerful tool for visualizing the sign of the expression in different intervals. We construct the table by listing the critical points in ascending order and then testing the sign of each factor (x+4) and (2x-1) in each interval.

In this table, we've chosen a test value within each interval and plugged it into the expressions (x+4) and (2x-1). The sign of the result is then recorded in the table. The sign of the entire expression (x+4)/(2x-1) is determined by the product of the signs of the numerator and the denominator.

Step 3: Identifying the Solution Intervals

We are looking for the intervals where the expression (x+4)/(2x-1) is less than zero, meaning it is negative. From our sign table, we can see that this occurs in the interval -4 < x < 1/2.

It is important to note that we use strict inequalities (less than) because the original inequality is strict. This means that we do not include the critical points themselves in the solution. Furthermore, we must exclude x = 1/2 because it makes the denominator zero, which is undefined.

Therefore, the solution to the inequality is -4 < x < 1/2.

Expressing the Solution: Different Notations

The solution to the inequality (x+4)/(2x-1) < 0 can be expressed in several ways, each with its own advantages. Let's explore the common notations:

1. Interval Notation

Interval notation is a concise way to represent a range of values using parentheses and brackets. Parentheses indicate that the endpoint is not included in the interval, while brackets indicate that it is included. In our case, the solution is expressed as (-4, 1/2). The parentheses signify that neither -4 nor 1/2 is included in the solution set.

2. Inequality Notation

Inequality notation uses inequality symbols (>, <, ≥, ≤) to define the range of values. For our solution, the inequality notation is -4 < x < 1/2. This directly states that 'x' is greater than -4 and less than 1/2.

3. Set-Builder Notation

Set-builder notation provides a more formal way to define the solution set using set theory. The solution can be written as {x | -4 < x < 1/2}. This reads as