Finding The Number Line Representation For The Inequality 3(8-4x) < 6(x-5)

Jul 12, 2025 by ADMIN 75 views

Solving the Inequality: A Step-by-Step Guide to Finding the Correct Number Line Representation

In the realm of mathematics, inequalities play a crucial role in defining relationships between values. Solving inequalities involves finding the range of values that satisfy the given condition. One common way to represent the solution set of an inequality is through a number line. This article delves into the process of solving the inequality $3(8-4x) < 6(x-5)$ and accurately depicting its solution set on a number line.

Understanding Inequalities

Before we delve into the specifics of solving the inequality, let's establish a firm grasp of what inequalities are. Unlike equations, which assert the equality of two expressions, inequalities express a relationship where two values are not necessarily equal. Inequalities use symbols like '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to) to indicate the relative order of values.

When solving inequalities, our objective is to isolate the variable on one side of the inequality sign. This process mirrors the steps involved in solving equations, with one crucial distinction: multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. This rule is paramount to ensure the accuracy of the solution.

Step-by-Step Solution of the Inequality

Now, let's embark on the journey of solving the inequality $3(8-4x) < 6(x-5)$ . We will meticulously follow each step to arrive at the solution set:

Distribute: Begin by distributing the constants on both sides of the inequality:
```
3 * 8 - 3 * 4x < 6 * x - 6 * 5
```
This simplifies to:
```
24 - 12x < 6x - 30
```
Combine Like Terms: Our next goal is to gather the 'x' terms on one side of the inequality and the constant terms on the other. Let's add 12x to both sides:
```
24 - 12x + 12x < 6x - 30 + 12x
```
This yields:
```
24 < 18x - 30
```
Now, let's add 30 to both sides:
```
24 + 30 < 18x - 30 + 30
```
This simplifies to:
```
54 < 18x
```
Isolate the Variable: To isolate 'x', we need to divide both sides of the inequality by 18:
```
54 / 18 < 18x / 18
```
This gives us:
```
3 < x
```
Rewrite the Inequality: For clarity and convention, we typically write the variable on the left side of the inequality. Therefore, we can rewrite the solution as:
```
x > 3
```

This solution signifies that any value of 'x' greater than 3 will satisfy the original inequality.

Representing the Solution Set on a Number Line

Now that we have determined the solution set, x > 3, let's visualize it on a number line. A number line is a graphical representation of numbers, extending infinitely in both positive and negative directions. To represent the solution set, we follow these steps:

Draw a Number Line: Begin by drawing a horizontal line. Mark the number 3 on this line. This point serves as our boundary.
Open Circle or Closed Circle: Since our solution is x > 3 (greater than, not greater than or equal to), we use an open circle at 3. An open circle indicates that 3 is not included in the solution set. If the inequality were x ≥ 3, we would use a closed circle to signify that 3 is included.
Shading the Solution Set: The inequality x > 3 means we are interested in all values greater than 3. Therefore, we shade the portion of the number line to the right of 3, extending towards positive infinity. This shaded region visually represents all the numbers that satisfy the inequality.

Interpreting the Number Line Representation

The number line representation provides a clear visual depiction of the solution set. It tells us that any number to the right of 3, excluding 3 itself, will make the original inequality true. For instance, 3.1, 4, 10, and 100 are all part of the solution set.

Conversely, any number less than or equal to 3 will not satisfy the inequality. Numbers like 2, 0, -5, and 3 are excluded from the solution set.

Common Mistakes to Avoid

When solving inequalities and representing their solutions on a number line, it's crucial to steer clear of common pitfalls. Here are some mistakes to watch out for:

Forgetting to Reverse the Inequality Sign: As mentioned earlier, multiplying or dividing both sides of an inequality by a negative number necessitates reversing the inequality sign. Failing to do so will lead to an incorrect solution.
Incorrectly Interpreting the Inequality Symbol: Be mindful of the difference between '<' (less than) and '≤' (less than or equal to), and similarly between '>' (greater than) and '≥' (greater than or equal to). The inclusion or exclusion of the boundary point depends on the inequality symbol used.
Shading the Wrong Region: Ensure you shade the correct portion of the number line based on the inequality. For x > a, shade to the right of 'a'; for x < a, shade to the left of 'a'.

Conclusion

Solving inequalities and representing their solutions on a number line is a fundamental skill in mathematics. By following a systematic approach, we can accurately determine the solution set and visualize it effectively. Remember to distribute, combine like terms, isolate the variable, and pay close attention to the direction of the inequality sign. With practice, you'll become adept at solving inequalities and confidently representing their solutions on number lines. The ability to accurately solve and represent inequalities is a valuable asset in various mathematical and real-world applications.

Number lines serve as a powerful tool in mathematics, particularly when it comes to visualizing inequalities. An inequality, unlike an equation, doesn't have a single solution but rather a range of values that satisfy the given condition. Representing these solutions graphically can be challenging, which is where number lines come into play. Number lines provide a clear and intuitive way to depict the solution set of an inequality, making it easier to understand and interpret.

Understanding Number Lines

A number line is essentially a straight line that represents all real numbers. It extends infinitely in both directions, with zero at the center, positive numbers to the right, and negative numbers to the left. Each point on the line corresponds to a unique real number. Number lines are a foundational concept in mathematics, used for various purposes such as ordering numbers, performing arithmetic operations, and, most importantly, visualizing inequalities.

When representing inequalities on a number line, we focus on the range of values that satisfy the inequality. This range, known as the solution set, is graphically depicted on the number line. The solution set may include all numbers greater than a certain value, all numbers less than a certain value, or a range of numbers between two values. The number line provides a visual representation of these possibilities.

Representing Solution Sets on Number Lines

The process of representing the solution set of an inequality on a number line involves several key steps. These steps ensure that the representation accurately reflects the solution.

Identify the Boundary Point: The first step is to identify the boundary point, which is the value at which the inequality transitions from being true to false or vice versa. This boundary point is typically determined by solving the inequality for the variable. For example, in the inequality x > 3, the boundary point is 3.
Use Open or Closed Circles: The next step is to represent the boundary point on the number line using either an open circle or a closed circle. The choice between these symbols depends on the inequality symbol used. If the inequality involves a strict inequality (< or >), we use an open circle to indicate that the boundary point is not included in the solution set. If the inequality involves a non-strict inequality (≤ or ≥), we use a closed circle to indicate that the boundary point is included in the solution set.
Shade the Solution Region: Once the boundary point is marked, we shade the region of the number line that represents the solution set. The direction of shading depends on the inequality symbol. If the inequality is of the form x > a, we shade to the right of the boundary point (a), indicating that all values greater than a satisfy the inequality. If the inequality is of the form x < a, we shade to the left of the boundary point (a), indicating that all values less than a satisfy the inequality.

Benefits of Using Number Lines to Visualize Inequalities

Using number lines to visualize inequalities offers several significant benefits. These benefits make number lines an indispensable tool for understanding and working with inequalities.

Clarity and Intuition: Number lines provide a clear and intuitive visual representation of the solution set. They allow us to see the range of values that satisfy the inequality, making it easier to grasp the concept.
Improved Understanding: Visualizing inequalities on a number line can enhance our understanding of the relationships between numbers and the concept of inequality itself. By seeing the solution set graphically, we can develop a deeper appreciation for the meaning of the inequality.
Problem-Solving Aid: Number lines can be a valuable aid in problem-solving. When faced with a complex inequality, representing the solution set on a number line can help us identify the correct range of values and avoid errors.
Communication Tool: Number lines serve as an effective communication tool. They allow us to clearly and concisely convey the solution set of an inequality to others.

Examples of Number Line Representations

Let's consider some examples to illustrate how number lines are used to represent the solution sets of inequalities.

x > 2: The solution set for this inequality includes all values greater than 2. On a number line, we would mark an open circle at 2 and shade the region to the right of 2.
x ≤ -1: The solution set for this inequality includes all values less than or equal to -1. On a number line, we would mark a closed circle at -1 and shade the region to the left of -1.
-3 < x < 5: This compound inequality represents the set of all values between -3 and 5, excluding -3 and 5. On a number line, we would mark open circles at -3 and 5 and shade the region between them.

Common Mistakes to Avoid When Using Number Lines

While number lines are a valuable tool, it's essential to avoid common mistakes when using them to represent inequalities. These mistakes can lead to misinterpretations of the solution set.

Incorrect Circle Type: Using the wrong type of circle (open or closed) can misrepresent the inclusion or exclusion of the boundary point. Remember to use an open circle for strict inequalities (< or >) and a closed circle for non-strict inequalities (≤ or ≥).
Incorrect Shading Direction: Shading the wrong region of the number line will result in an inaccurate representation of the solution set. Ensure you shade to the right for inequalities of the form x > a and to the left for inequalities of the form x < a.
Misinterpreting Compound Inequalities: Compound inequalities, which involve two inequalities connected by