Solving The Inequality X(x-2)≥3: A Step-by-Step Guide

# Solving the Inequality x(x-2)≥3: A Comprehensive Guide

In the realm of mathematics, inequalities play a crucial role in defining ranges and constraints. Understanding how to solve inequalities is fundamental for various mathematical applications, from basic algebra to advanced calculus. This article will provide a detailed, step-by-step guide on solving the inequality x(x-2)≥3. We will break down the process into manageable steps, ensuring clarity and comprehension for readers of all levels. Whether you're a student grappling with homework or simply looking to refresh your mathematical skills, this guide will equip you with the knowledge and confidence to tackle similar problems. 

## Understanding Inequalities

Before we dive into the specifics of solving x(x-2)≥3, it's essential to grasp the basics of inequalities. Unlike equations, which seek to find specific values that make two expressions equal, inequalities deal with relationships where one expression is greater than, less than, greater than or equal to, or less than or equal to another. The symbols representing these relationships are >, <, ≥, and ≤, respectively. When dealing with inequalities, we're essentially identifying a range of values that satisfy the given condition, rather than a single solution. This range can be represented graphically on a number line or expressed in interval notation. The principles of solving inequalities share some similarities with solving equations, but there are key differences to be aware of, particularly when it comes to multiplying or dividing by negative numbers, which we'll explore later in this guide.

## Step-by-Step Solution

### 1. Expand and Rearrange the Inequality

The first step in solving the inequality x(x-2)≥3 is to expand the expression on the left side and rearrange the inequality to have zero on one side. This simplifies the problem and allows us to work with a more standard form. By expanding the left side, we get x^2 - 2x. Now, subtracting 3 from both sides of the inequality, we obtain the quadratic inequality x^2 - 2x - 3 ≥ 0. This form is crucial because it allows us to identify the critical points where the expression equals zero, which are essential for determining the intervals where the inequality holds true.

### 2. Factor the Quadratic Expression

Factoring the quadratic expression is a key step in finding the critical points. We need to find two numbers that multiply to -3 and add up to -2. These numbers are -3 and 1. Therefore, we can factor the quadratic expression as (x - 3)(x + 1) ≥ 0. Factoring the quadratic expression transforms the inequality into a product of two linear factors, making it easier to analyze the sign of the expression over different intervals. The factors directly reveal the roots of the quadratic equation, which are the critical points that divide the number line into intervals where the inequality's sign remains constant.

### 3. Identify Critical Points

The critical points are the values of x that make the expression equal to zero. In this case, the critical points are the solutions to the equation (x - 3)(x + 1) = 0. Setting each factor equal to zero, we find x = 3 and x = -1. These critical points are crucial because they divide the number line into intervals where the expression (x - 3)(x + 1) has a consistent sign (either positive or negative). The critical points are the boundaries of these intervals, and they play a vital role in determining the solution set of the inequality.

### 4. Create a Sign Chart

A sign chart is a visual tool that helps us determine the sign of the expression (x - 3)(x + 1) in the intervals defined by the critical points. Draw a number line and mark the critical points, -1 and 3. These points divide the number line into three intervals: (-∞, -1), (-1, 3), and (3, ∞). Now, we need to test a value from each interval in the expression (x - 3)(x + 1) to determine its sign in that interval. For example, in the interval (-∞, -1), we can test x = -2. The expression becomes (-2 - 3)(-2 + 1) = (-5)(-1) = 5, which is positive. In the interval (-1, 3), we can test x = 0. The expression becomes (0 - 3)(0 + 1) = (-3)(1) = -3, which is negative. In the interval (3, ∞), we can test x = 4. The expression becomes (4 - 3)(4 + 1) = (1)(5) = 5, which is positive. The sign chart visually represents the sign of the expression in each interval, making it easier to identify the intervals where the inequality holds true.

### 5. Determine the Solution Set

We are looking for the intervals where (x - 3)(x + 1) ≥ 0, which means the expression is either positive or equal to zero. From the sign chart, we found that the expression is positive in the intervals (-∞, -1) and (3, ∞). Additionally, the expression is equal to zero at the critical points x = -1 and x = 3. Therefore, the solution set includes these critical points and the intervals where the expression is positive. In interval notation, the solution set is (-∞, -1] ∪ [3, ∞). This notation indicates that the solution includes all values of x less than or equal to -1 and all values of x greater than or equal to 3.

## Alternative Method: Graphing the Quadratic

An alternative method to solving quadratic inequalities is by graphing the quadratic function. This approach offers a visual understanding of the solution set. The graph of the quadratic function y = x^2 - 2x - 3 is a parabola. The critical points, x = -1 and x = 3, are the x-intercepts of the parabola. The parabola opens upwards because the coefficient of the x^2 term is positive. The inequality x^2 - 2x - 3 ≥ 0 asks for the values of x where the parabola is above or on the x-axis. From the graph, we can see that the parabola is above the x-axis for x < -1 and x > 3, and it intersects the x-axis at x = -1 and x = 3. Therefore, the solution set is (-∞, -1] ∪ [3, ∞), which confirms our previous result obtained using the sign chart method. The graphical method provides a visual confirmation of the solution and can be particularly helpful for those who prefer a geometric approach.

## Common Mistakes to Avoid

When solving inequalities, it's crucial to avoid common mistakes that can lead to incorrect solutions. One frequent error is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. For example, if you have -2x > 4, dividing both sides by -2 requires flipping the inequality sign, resulting in x < -2. Another common mistake is incorrectly interpreting the solution set. It's essential to pay close attention to the inequality sign (≥, ≤, >, <) and include or exclude the critical points accordingly. For instance, if the inequality is strict (e.g., > or <), the critical points are not included in the solution set, and we use parentheses in interval notation. Additionally, students sometimes make errors when factoring the quadratic expression or when constructing the sign chart. Careful attention to detail and thorough checking of each step can help prevent these mistakes. Regular practice and review of the fundamental principles of inequalities are essential for mastering this topic.

## Real-World Applications

Inequalities are not just abstract mathematical concepts; they have numerous real-world applications in various fields. In economics, inequalities are used to model constraints such as budget limitations or resource availability. For example, a consumer's budget constraint can be expressed as an inequality that limits the amount they can spend on different goods. In physics, inequalities are used to describe physical limitations, such as the maximum speed of an object or the range of a projectile's motion. In engineering, inequalities are used in optimization problems to design structures that can withstand certain loads or to minimize costs while meeting performance requirements. For instance, an engineer might use inequalities to ensure that the stress on a bridge does not exceed a certain threshold. Inequalities also play a crucial role in computer science, particularly in algorithm analysis and optimization. Understanding how to solve inequalities is therefore not only essential for academic success but also for solving practical problems in various professional domains.

## Conclusion

Solving the inequality x(x-2)≥3 involves a series of steps, from expanding and rearranging the expression to identifying critical points and determining the solution set. This guide has provided a comprehensive walkthrough of the process, highlighting key concepts and potential pitfalls along the way. By understanding the principles behind solving inequalities and practicing regularly, you can develop the skills and confidence to tackle a wide range of mathematical problems. Remember, mathematics is a journey of continuous learning and exploration. Each problem you solve is a step forward in your understanding of the world around you.