Solving Compound Inequalities A Step-by-Step Guide

Understanding Compound Inequalities

Before we dive into the solution, let's first understand what compound inequalities are. Compound inequalities are mathematical statements that combine two or more inequalities using the words "and" or "or." The inequality we're going to solve, $3 extless= \frac{1}{2} x+2 < 5$, is a type of compound inequality that uses "and" implicitly. It means that the expression $\frac{1}{2} x+2$ must be greater than or equal to 3 and less than 5. In essence, we are looking for the values of x that satisfy both conditions simultaneously. To solve this, we will isolate x by performing algebraic operations on all parts of the inequality, ensuring that we maintain the balance and integrity of the mathematical statement. Compound inequalities are not just abstract mathematical concepts; they have practical applications in various fields, including economics, engineering, and computer science. Understanding how to solve them is a valuable skill for anyone pursuing studies or careers in these areas. The ability to manipulate and solve inequalities allows for the modeling and analysis of real-world constraints and conditions, making it an essential tool in problem-solving and decision-making.

Step-by-Step Solution

1. Isolate the Term with x

Our goal is to isolate x in the middle of the compound inequality. To begin, we need to eliminate the constant term, which is +2. We can do this by subtracting 2 from all three parts of the inequality:

$3 \leq \frac{1}{2} x+2 < 5$ becomes:

$3 - 2 \leq \frac{1}{2} x+2 - 2 < 5 - 2$

Simplifying this gives us:

$1 \leq \frac{1}{2} x < 3$

This step is crucial because it simplifies the inequality, bringing us closer to isolating x. By performing the same operation on all parts of the inequality, we maintain the balance and ensure that the solution set remains unchanged. The subtraction property of inequality allows us to manipulate the inequality without altering the fundamental relationship between the expressions. This is a key principle in solving inequalities and ensures that our solution remains accurate and valid. Mastering this step is essential for tackling more complex compound inequalities and other algebraic problems.

2. Eliminate the Fraction

Now, we have $1 \leq \frac{1}{2} x < 3$. To isolate x completely, we need to get rid of the fraction $\frac{1}{2}$. We can do this by multiplying all three parts of the inequality by 2:

$2 * 1 \leq 2 * \frac{1}{2} x < 2 * 3$

This simplifies to:

$2 \leq x < 6$

By multiplying all parts of the inequality by 2, we effectively cancel out the fraction, leaving x by itself in the middle. This multiplication is based on the multiplication property of inequality, which states that multiplying all sides of an inequality by a positive number preserves the inequality. This step is a critical maneuver in solving inequalities involving fractions, and it's a technique that is widely applicable in various mathematical contexts. Understanding how to manipulate inequalities with fractions is an essential skill for anyone working with algebraic expressions and equations. The ability to quickly and accurately eliminate fractions simplifies the problem and makes it easier to identify the solution set.

3. Interpret the Solution

The solution $2 \leq x < 6$ tells us that x is greater than or equal to 2 and less than 6. This means that x can be any number within this range, including 2 but not including 6. The inequality includes 2 because of the "less than or equal to" symbol ($\leq$), but it does not include 6 because of the "less than" symbol (<). Understanding the nuances of these inequality symbols is crucial for correctly interpreting and representing the solution set. The solution set represents all the values of x that satisfy the original compound inequality. In practical terms, this could represent a range of acceptable values in a real-world application, such as acceptable temperatures for a chemical reaction or the range of inputs for a control system. The ability to interpret inequalities and understand their practical implications is a valuable skill in many fields.

4. Express the Solution in Interval Notation

To express the solution in interval notation, we use brackets and parentheses. A bracket [ ] indicates that the endpoint is included in the solution, while a parenthesis ( ) indicates that the endpoint is not included. For the inequality $2 \leq x < 6$, the interval notation is:

$[2, 6)$

The square bracket [ on the left side indicates that 2 is included in the solution set, while the parenthesis ) on the right side indicates that 6 is not included. Interval notation is a concise and standard way of representing the solution set of inequalities, and it's widely used in mathematics and related fields. Understanding how to convert inequalities into interval notation is essential for clear communication and accurate representation of mathematical solutions. Interval notation not only simplifies the expression of solution sets but also facilitates operations on intervals, such as unions and intersections, which are fundamental concepts in set theory and analysis.

Representing the Solution Graphically

Visualizing the solution on a number line can provide a deeper understanding of the solution set. The solution $2 \leq x < 6$ can be represented graphically by drawing a number line and marking the interval between 2 and 6. A closed circle (or a bracket) is used at 2 to indicate that it is included in the solution, and an open circle (or a parenthesis) is used at 6 to indicate that it is not included. The portion of the number line between 2 and 6 is shaded to represent all the values of x that satisfy the inequality. Graphical representation of solutions is a powerful tool for visualizing and understanding mathematical concepts, especially in the context of inequalities and intervals. It provides a clear and intuitive way to see the range of values that satisfy the given conditions. This visual representation can be particularly helpful for students who are learning about inequalities and for anyone who wants to quickly grasp the solution set.

Conclusion

The solution set in interval notation for the compound inequality $3 \leq \frac{1}{2} x+2 < 5$ is $[2, 6)$. We arrived at this solution by systematically isolating x and then expressing the result in the appropriate notation. Solving compound inequalities is a critical skill in algebra, with applications in various fields. By mastering the steps outlined in this guide, you can confidently tackle similar problems and enhance your understanding of mathematical inequalities. The ability to solve compound inequalities is not just a theoretical exercise; it's a practical skill that is valuable in a wide range of real-world applications, from finance and economics to engineering and computer science. Whether you're a student, a professional, or simply someone who enjoys mathematical challenges, understanding and applying these techniques will enhance your problem-solving abilities and broaden your mathematical horizons. The journey of mastering mathematical concepts is a continuous one, and each step forward contributes to a deeper and more comprehensive understanding of the world around us.