The Ultimate Inequality Puzzle: Decoding the Number Line for 3(8 – 4x) < 6(X – 5)
When it comes to solving inequalities, visual representation can often provide a clearer understanding of the solution set. In this case, we are tasked with finding which number line represents the solution set for the inequality 3(8 – 4x) < 6(X – 5). By examining the given inequality and analyzing the expressions within it, we can determine the appropriate number line that accurately represents the values that satisfy the inequality. So, let's embark on this mathematical journey and discover the most fitting number line for our solution set.
Introduction
In this article, we will discuss which number line represents the solution set for the inequality 3(8 – 4x) < 6(X – 5). We will go through the steps required to solve the inequality and graph the solution on a number line. By the end of this article, you will have a clear understanding of how to represent the solution set visually.
Simplifying the Inequality
The first step in solving the inequality is to simplify it. Let's start by distributing the multiplication on both sides of the equation:
24 - 12x < 6x - 30
Combining Like Terms
Now, let's combine like terms on both sides of the inequality:
-12x - 6x < -30 - 24
-18x < -54
Dividing by a Negative Number
Since we have a negative coefficient (-18x), we need to divide both sides of the inequality by -18. However, when dividing by a negative number, the inequality symbol needs to be flipped:
x > -54 / -18
x > 3
Graphing the Solution Set
Now that we have found the value of x, let's graph the solution set on a number line. We will represent all values of x greater than 3.
Locating the Origin
First, let's locate the origin (0) on the number line. This will be our starting point for graphing the solution set.
Marking the Value of 3
Next, we will mark the value of 3 on the number line. Since the inequality is x > 3, the point representing 3 will be an open circle.
Shading the Solution Set
Finally, we will shade the region to the right of the open circle, representing all values of x greater than 3. This shaded region represents the solution set for the inequality.
Conclusion
In conclusion, the number line that represents the solution set for the inequality 3(8 – 4x) < 6(X – 5) is a number line with an open circle at 3 and a shaded region to the right of 3. This indicates that all values of x greater than 3 satisfy the inequality. By following the steps outlined in this article, you can effectively solve and graph similar inequalities on a number line.
Introduction: Understanding the inequality and its solution set
Inequalities are mathematical expressions that compare two values, indicating whether one is greater than or less than the other. The solution set for an inequality represents the range of values that satisfy the given inequality. In this case, we will be analyzing the inequality 3(8 – 4x) < 6(X – 5) to determine the number line that represents its solution set.
Simplifying the inequality: Transform numerical expressions to make the equation easier to solve
To simplify the given inequality, we need to apply various algebraic techniques to eliminate any unnecessary elements. This involves transforming the numerical expressions by performing operations such as addition, subtraction, multiplication, and division.
Distributive property: Applying the distributive property to eliminate parentheses in the inequality
The distributive property allows us to distribute a factor to each term inside parentheses. In this case, we can distribute the factor 3 to both terms in the expression (8 – 4x), resulting in 24 – 12x. Similarly, distributing the factor 6 to (X – 5) gives us 6X – 30.
Combining like terms: Simplifying both sides of the inequality by combining similar terms
After applying the distributive property, we combine like terms on both sides of the inequality. This involves adding or subtracting terms that have the same variable and exponent. In our case, we have -12x on the left side and 6X on the right side. Combining these terms gives us -12x - 6X.
Isolating the variable: Rearranging the inequality to have the variable on one side and the constant on the other
To solve the inequality, we need to isolate the variable on one side of the inequality symbol and the constant term on the other side. In our case, we can achieve this by moving the -6X term to the left side of the inequality. This gives us -12x - 6X < 24 – 30.
Addition and subtraction: Determining the appropriate number line to represent the solution set involving addition and subtraction
The inequality involves addition and subtraction operations, which require a number line to represent the solution set accurately. To determine the appropriate number line, we need to consider the direction of the inequality symbol and whether the solution is inclusive or exclusive of certain values.
Multiplication and division: Determining the appropriate number line to represent the solution set involving multiplication and division
In some cases, inequalities involve multiplication and division operations. These operations also require a number line to represent the solution set. The direction of the inequality symbol and the properties of multiplication and division will guide us in determining the correct number line representation.
Solving the inequality: Plotting the solution set on the number line using open or closed circles
Once we have determined the appropriate number line, we can plot the solution set for the inequality. The solution set consists of all the values that satisfy the given inequality. Depending on the inclusivity of the solution, we use open or closed circles on the number line to represent the values.
Interval notation: Expressing the solution set using interval notation to represent the range of values that satisfy the inequality
Interval notation is a concise and efficient way to express the solution set of an inequality. It provides a range of values that satisfy the inequality by using brackets or parentheses to denote inclusivity or exclusivity of the endpoints. By converting the solution set into interval notation, we can express it in a more compact and standardized format.
Checking the solution: Verifying if the solution set obtained satisfies the original inequality and discussing any restrictions or special cases
After obtaining the solution set and expressing it in interval notation, it is crucial to check if the values in the solution set satisfy the original inequality. This step ensures the accuracy of our solution and helps identify any restrictions or special cases that may arise. By verifying the solution and considering any limitations, we can confidently conclude the analysis of the given inequality.
Which Number Line Represents The Solution Set For The Inequality 3(8 – 4x) < 6(X – 5)?
Explanation of the Inequality:
The given inequality is: 3(8 – 4x) < 6(X – 5).
To find the solution set for this inequality, we need to simplify and solve for x.
Simplifying the Inequality:
- Distribute the 3 on the left side of the inequality: 24 - 12x < 6(X - 5).
- Expand the right side of the inequality: 24 - 12x < 6X - 30.
- Combine like terms on both sides: 24 + 30 < 6X + 12x.
- Simplify further: 54 < 18x.
- Divide both sides by 18 to isolate x: 54/18 < 18x/18.
- Simplify: 3 < x.
Solution Set:
The solution set for the inequality 3(8 – 4x) < 6(X – 5) is represented by the number line where x is greater than 3. This means that any value of x greater than 3 will satisfy the inequality.
Number Line Representation:
On the number line, we represent the solution set by marking a point at 3 and shading everything to the right of it. Any value of x that falls within this shaded region will make the inequality true. Values less than 3 are not included in the solution set.
x | Inequality | Solution |
---|---|---|
2 | 3(8 - 4x) < 6(X - 5) | False |
3 | 3(8 - 4x) < 6(X - 5) | False |
4 | 3(8 - 4x) < 6(X - 5) | True |
5 | 3(8 - 4x) < 6(X - 5) | True |
6 | 3(8 - 4x) < 6(X - 5) | True |
Thank you for taking the time to visit our blog and read about the solution set for the inequality 3(8 – 4x) < 6(X – 5). We hope that this article has provided you with a clear understanding of how to represent the solution set on a number line. In this closing message, we will briefly summarize the key points discussed in the article and emphasize the importance of understanding inequalities in mathematics.
In the article, we explored the process of solving the inequality 3(8 – 4x) < 6(X – 5) step by step. We began by simplifying both sides of the inequality and distributing the coefficients. This allowed us to eliminate the parentheses and combine like terms. By rearranging the terms and isolating the variable on one side, we obtained an equivalent inequality in the form of ax + b < cx + d.
Using this form, we were able to identify the coefficients a, b, c, and d to plot the solution set on a number line. We learned that the solution set represents all the values of x that satisfy the given inequality. By shading the appropriate interval on the number line, we visually represented the solution set.
Understanding inequalities is crucial in various areas of mathematics and real-life applications. It allows us to analyze relationships between quantities and make informed decisions based on the constraints they impose. Whether you are studying algebra, calculus, or other mathematical subjects, having a solid understanding of inequalities will enhance your problem-solving skills and enable you to tackle more complex mathematical concepts.
We hope that this article has been helpful in clarifying the representation of the solution set for the inequality 3(8 – 4x) < 6(X – 5). If you have any further questions or need additional assistance, please feel free to reach out to us. Thank you again for visiting our blog, and we look forward to providing you with more informative content in the future.
Which Number Line Represents The Solution Set For The Inequality 3(8 – 4x) < 6(X – 5)?
1. What is the given inequality?
The given inequality is 3(8 – 4x) < 6(X – 5).
2. How can we solve this inequality?
To solve this inequality, we need to simplify it and isolate the variable.
- Distribute the 3 and 6 on both sides of the inequality: 24 - 12x < 6x - 30.
- Combine like terms: -12x - 6x < -30 - 24.
- Simplify further: -18x < -54.
- Divide both sides by -18, but remember to flip the inequality sign since we are dividing by a negative number: x > 3.
3. What does the solution set represent?
The solution set represents all the values of x that satisfy the given inequality. In this case, x is greater than 3.
4. How can we represent the solution set on a number line?
On a number line, we can represent the solution set by marking a point at 3 (since x is greater than 3) and shading the region to the right of that point. Any value of x in that shaded region will satisfy the given inequality.
Example:
Number line:
-∞ -2 -1 0 1 2 3 4 5 6 7 8 9 10 ∞ ------------------------------------------------------ x x x x x x x x x x
In the above example, the shaded region to the right of 3 represents the solution set for the inequality x > 3.