Solving Inequality -2 ≤ 2x + 1/2 - (x - 2/3) ≤ 4 With Number Line Illustration

Introduction to Inequalities

In mathematics, inequalities play a crucial role in defining ranges and intervals within which solutions to equations may lie. Unlike equations that seek specific values, inequalities describe relationships where one expression is greater than, less than, greater than or equal to, or less than or equal to another. This article delves into the process of solving a specific compound inequality, namely $-2 \leq 2x + \frac{1}{2} - (x - \frac{2}{3}) \leq 4$, and subsequently illustrates the solution set on a number line. Understanding inequalities is fundamental not only in algebra but also in various fields like calculus, optimization, and real-world problem-solving where constraints and bounds are essential considerations. This detailed exploration will enhance your ability to tackle similar problems and appreciate the practical applications of mathematical inequalities.

Understanding Compound Inequalities

Before we delve into solving the given inequality, it's essential to grasp the concept of compound inequalities. A compound inequality is essentially two or more inequalities that are combined into a single statement. These inequalities are typically connected by the words "and" or "or." In our case, we have a compound inequality that uses the "and" condition implicitly, as it specifies that the expression $2x + \frac{1}{2} - (x - \frac{2}{3})$ must simultaneously be greater than or equal to $-2$ and less than or equal to $4$. Such inequalities define a range or interval of values for the variable $x$ that satisfy both conditions. This type of inequality is quite common in mathematical analysis and is used extensively in fields that require defining constraints, such as linear programming and optimization problems. The key to solving compound inequalities lies in treating each part of the inequality separately while maintaining the logical connection between them, ensuring that the final solution satisfies all conditions.

Breaking Down the Compound Inequality

The compound inequality $-2 \leq 2x + \frac{1}{2} - (x - \frac{2}{3}) \leq 4$ can be seen as two separate inequalities connected by an implicit "and":

1. $-2 \leq 2x + \frac{1}{2} - (x - \frac{2}{3})$
2. $2x + \frac{1}{2} - (x - \frac{2}{3}) \leq 4$

To solve this compound inequality, we must solve each inequality individually and then find the intersection of their solution sets. This means we are looking for the values of $x$ that satisfy both inequalities simultaneously. This approach is crucial because it ensures that the final solution adheres to all the constraints defined by the original compound inequality. Breaking down the complex inequality into simpler parts makes the problem more manageable and reduces the chances of errors. Each inequality will be simplified using algebraic manipulations, and then the solutions will be combined to provide the complete solution set.

Step-by-Step Solution

Let's proceed with the step-by-step solution of the inequality. Our primary goal is to isolate $x$ in the middle of the compound inequality. To achieve this, we will perform algebraic manipulations on all parts of the inequality, ensuring that we maintain the balance and validity of the inequality throughout the process. This methodical approach is essential to accurately determine the range of values that satisfy the given conditions. We will start by simplifying the expression within the inequality, then proceed with adding or subtracting constants, and finally, multiply or divide to isolate $x$. Each step will be carefully explained to ensure clarity and understanding of the solution process.

1. Simplify the Expression

The first step is to simplify the expression $2x + \frac{1}{2} - (x - \frac{2}{3})$. We begin by distributing the negative sign:

$2x + \frac{1}{2} - x + \frac{2}{3}$

Next, combine like terms:

$(2x - x) + (\frac{1}{2} + \frac{2}{3})$

This simplifies to:

$x + \frac{1}{2} + \frac{2}{3}$

To add the fractions, we find a common denominator, which is 6:

$x + \frac{3}{6} + \frac{4}{6}$

Combining the fractions gives:

$x + \frac{7}{6}$

So, our compound inequality now looks like this:

$-2 \leq x + \frac{7}{6} \leq 4$

2. Isolate $x$

To isolate $x$, we need to subtract $\frac{7}{6}$ from all parts of the inequality:

$-2 - \frac{7}{6} \leq x + \frac{7}{6} - \frac{7}{6} \leq 4 - \frac{7}{6}$

This simplifies to:

$-2 - \frac{7}{6} \leq x \leq 4 - \frac{7}{6}$

Now, we need to perform the subtractions. First, convert the integers to fractions with a denominator of 6:

$\frac{-12}{6} - \frac{7}{6} \leq x \leq \frac{24}{6} - \frac{7}{6}$

Perform the subtractions:

$\frac{-19}{6} \leq x \leq \frac{17}{6}$

Thus, the solution to the inequality is:

$-\frac{19}{6} \leq x \leq \frac{17}{6}$

This means that $x$ can be any value between $-\frac{19}{6}$ and $\frac{17}{6}$, inclusive. This range of values satisfies the original compound inequality. The next step is to represent this solution graphically on a number line, providing a visual representation of the solution set.

Illustrating the Solution on a Number Line

Visualizing the solution on a number line is an effective way to understand the range of values that satisfy the inequality. A number line is a graphical representation of real numbers, extending infinitely in both positive and negative directions. To illustrate our solution $-\frac{19}{6} \leq x \leq \frac{17}{6}$, we will mark the endpoints of the interval on the number line and shade the region between them. The endpoints, $-\frac{19}{6}$ and $\frac{17}{6}$, are crucial as they define the boundaries of the solution set. The use of closed circles or brackets at these points indicates that the endpoints are included in the solution, reflecting the "less than or equal to" and "greater than or equal to" conditions in our inequality. This visual representation not only clarifies the solution set but also aids in understanding the concept of intervals and how inequalities define specific ranges of values on the real number line.

Steps to Illustrate the Solution

1. Draw a Number Line: Start by drawing a straight line and marking zero as a reference point. Extend the line in both positive and negative directions.
2. Mark the Endpoints: Locate $-\frac{19}{6}$ and $\frac{17}{6}$ on the number line. Note that $-\frac{19}{6}$ is approximately $-3.17$, and $\frac{17}{6}$ is approximately $2.83$. Place these values on the number line relative to zero.
3. Use Closed Circles or Brackets: Since the inequality includes "equal to" ($\leq$), we use closed circles (filled-in circles) or square brackets at the endpoints. This indicates that $-\frac{19}{6}$ and $\frac{17}{6}$ are included in the solution set.
4. Shade the Region: Shade the region between $-\frac{19}{6}$ and $\frac{17}{6}$ on the number line. This shaded region represents all the values of $x$ that satisfy the inequality. Any point within this region, including the endpoints, is a valid solution.

Visual Representation

Imagine a number line where the segment from approximately -3.17 to 2.83 is shaded, with closed circles at both ends. This shaded segment visually represents the solution set of the inequality $-\frac{19}{6} \leq x \leq \frac{17}{6}$. This visual aid is particularly helpful in understanding the continuous nature of the solution set and how the inequality defines a range of permissible values for $x$. The number line representation provides a clear and intuitive understanding of the solution, making it easier to grasp the concept of inequalities and their solutions.

Conclusion

In conclusion, we have successfully solved the compound inequality $-2 \leq 2x + \frac{1}{2} - (x - \frac{2}{3}) \leq 4$ and found the solution to be $-\frac{19}{6} \leq x \leq \frac{17}{6}$. This means that any value of $x$ within this interval, including the endpoints, satisfies the original inequality. The step-by-step solution involved simplifying the expression, isolating $x$, and expressing the solution in its simplest form. Furthermore, we illustrated the solution on a number line, providing a visual representation of the range of values that $x$ can take. This graphical representation is a powerful tool for understanding inequalities and their solutions, as it clearly shows the continuous nature of the solution set.

Understanding and solving inequalities is a fundamental skill in mathematics with wide-ranging applications. From determining feasible regions in linear programming to defining constraints in optimization problems, inequalities are essential for modeling real-world scenarios. The ability to manipulate and solve inequalities, as demonstrated in this article, equips you with a valuable tool for problem-solving in various contexts. By mastering these techniques, you can confidently tackle more complex mathematical problems and apply these skills in practical situations.