Which Number Line Represents The Solution To $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$?
Introduction to Inequalities and Number Lines
In mathematics, inequalities are used to compare two or more values. They are essential in solving various problems, including those involving number lines. A number line is a visual representation of numbers on a line, allowing us to graphically represent the solution to an inequality. In this article, we will explore how to represent the solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$ on a number line.
Understanding the Inequality
The given inequality is $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$. To solve this inequality, we need to isolate the variable x. We can start by adding 1.2x to both sides of the inequality, which gives us:
Next, we can subtract 6.5 from both sides of the inequality, resulting in:
Solving for x
To solve for x, we need to isolate the variable. We can do this by dividing both sides of the inequality by -2, which gives us:
However, when we divide by a negative number, the inequality sign is reversed. Therefore, the correct solution is:
Graphing the Solution on a Number Line
Now that we have the solution to the inequality, we can graph it on a number line. A number line is a visual representation of numbers on a line, allowing us to graphically represent the solution to an inequality. To graph the solution, we need to plot a point on the number line that represents the solution.
In this case, the solution is x < 2. We can plot a point at x = 2, which is the boundary of the solution. Since the solution is less than 2, we can shade the region to the left of the point to represent the solution.
Conclusion
In conclusion, the number line that represents the solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$ is a number line with a point at x = 2, and the region to the left of the point shaded. This represents the solution x < 2.
Frequently Asked Questions
- Q: What is the solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$? A: The solution to the inequality is x < 2.
- Q: How do I graph the solution on a number line? A: To graph the solution, plot a point at x = 2, and shade the region to the left of the point to represent the solution.
- Q: What is the boundary of the solution? A: The boundary of the solution is x = 2.
Step-by-Step Solution
- Start by adding 1.2x to both sides of the inequality.
- Subtract 6.5 from both sides of the inequality.
- Divide both sides of the inequality by -2.
- Plot a point at x = 2 on the number line.
- Shade the region to the left of the point to represent the solution.
Common Mistakes to Avoid
- When dividing by a negative number, the inequality sign is reversed.
- When graphing the solution on a number line, make sure to plot the point at the boundary of the solution.
- When shading the region, make sure to shade the region to the left of the point to represent the solution.
Real-World Applications
Inequalities and number lines have many real-world applications. For example, in finance, inequalities are used to compare the value of investments. In engineering, inequalities are used to design and optimize systems. In medicine, inequalities are used to compare the effectiveness of treatments.
Final Thoughts
In conclusion, the number line that represents the solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$ is a number line with a point at x = 2, and the region to the left of the point shaded. This represents the solution x < 2. We hope this article has provided a clear understanding of how to represent the solution to an inequality on a number line.
Introduction
In our previous article, we explored how to represent the solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$ on a number line. In this article, we will answer some of the most frequently asked questions about inequalities and number lines.
Q&A
Q: What is the difference between an inequality and an equation?
A: An inequality is a statement that compares two or more values, while an equation is a statement that states that two or more values are equal.
Q: How do I solve an inequality?
A: To solve an inequality, you need to isolate the variable by performing operations on both sides of the inequality.
Q: What is the boundary of the solution?
A: The boundary of the solution is the value of the variable that separates the solution from the non-solution.
Q: How do I graph the solution on a number line?
A: To graph the solution, plot a point at the boundary of the solution, and shade the region to the left or right of the point to represent the solution.
Q: What is the significance of the inequality sign?
A: The inequality sign indicates the direction of the solution. For example, if the inequality sign is <, the solution is less than the boundary.
Q: Can I use a number line to solve a system of inequalities?
A: Yes, you can use a number line to solve a system of inequalities. However, you need to graph each inequality separately and then find the intersection of the solutions.
Q: How do I determine the direction of the solution on a number line?
A: To determine the direction of the solution, look at the inequality sign. If the inequality sign is <, the solution is to the left of the boundary. If the inequality sign is >, the solution is to the right of the boundary.
Q: Can I use a number line to solve a rational inequality?
A: Yes, you can use a number line to solve a rational inequality. However, you need to graph each rational function separately and then find the intersection of the solutions.
Q: How do I graph a rational function on a number line?
A: To graph a rational function on a number line, plot the points where the function is undefined, and then graph the function by connecting the points.
Q: What is the significance of the asymptotes in a rational function?
A: The asymptotes in a rational function indicate the behavior of the function as x approaches positive or negative infinity.
Q: Can I use a number line to solve a quadratic inequality?
A: Yes, you can use a number line to solve a quadratic inequality. However, you need to graph the quadratic function and then find the intersection of the solutions.
Q: How do I graph a quadratic function on a number line?
A: To graph a quadratic function on a number line, plot the vertex of the parabola, and then graph the function by connecting the points.
Q: What is the significance of the vertex in a quadratic function?
A: The vertex in a quadratic function indicates the minimum or maximum value of the function.
Conclusion
In conclusion, inequalities and number lines are powerful tools for solving mathematical problems. By understanding how to represent the solution to an inequality on a number line, you can solve a wide range of mathematical problems. We hope this article has provided a clear understanding of the frequently asked questions about inequalities and number lines.
Final Thoughts
Inequalities and number lines have many real-world applications. For example, in finance, inequalities are used to compare the value of investments. In engineering, inequalities are used to design and optimize systems. In medicine, inequalities are used to compare the effectiveness of treatments.
Step-by-Step Solution
- Start by understanding the inequality and the number line.
- Solve the inequality by isolating the variable.
- Graph the solution on a number line by plotting a point at the boundary of the solution.
- Shade the region to the left or right of the point to represent the solution.
- Determine the direction of the solution by looking at the inequality sign.
Common Mistakes to Avoid
- When solving an inequality, make sure to isolate the variable by performing operations on both sides of the inequality.
- When graphing the solution on a number line, make sure to plot a point at the boundary of the solution.
- When shading the region, make sure to shade the region to the left or right of the point to represent the solution.
Real-World Applications
Inequalities and number lines have many real-world applications. For example, in finance, inequalities are used to compare the value of investments. In engineering, inequalities are used to design and optimize systems. In medicine, inequalities are used to compare the effectiveness of treatments.
Final Thoughts
In conclusion, inequalities and number lines are powerful tools for solving mathematical problems. By understanding how to represent the solution to an inequality on a number line, you can solve a wide range of mathematical problems. We hope this article has provided a clear understanding of the frequently asked questions about inequalities and number lines.