Does The Point \[$(1, 2)\$\] Satisfy The Inequality \[$6x + 5y \ \textless \ 19?\$\]A. Yes B. No
Introduction
In mathematics, inequalities are used to describe relationships between variables. They are essential in various fields, including algebra, geometry, and calculus. In this article, we will discuss whether a given point satisfies a specific inequality. We will use the point {(1, 2)$}$ and the inequality ${6x + 5y \ \textless \ 19?\$} to determine if the point lies within the region defined by the inequality.
Understanding the Inequality
The given inequality is ${6x + 5y \ \textless \ 19?\$}. This is a linear inequality, which means it can be represented graphically as a line on a coordinate plane. The inequality states that the sum of ${6x\$} and ${5y\$} is less than ${19\$}. To determine if the point {(1, 2)$}$ satisfies this inequality, we need to substitute the values of {x$}$ and {y$}$ into the inequality and evaluate the result.
Substituting the Values
To substitute the values of {x$}$ and {y$}$ into the inequality, we replace {x$}$ with ${1\$} and {y$}$ with ${2\$}. This gives us:
${$6(1) + 5(2) \ \textless \ 19?$
Evaluating the Expression
Now, we need to evaluate the expression [6(1) + 5(2)\$}. This involves multiplying ${6\$} by ${1\$} and ${5\$} by ${2\$}, and then adding the results.
${$6(1) = 6$ [$5(2) = 10$ [$6 + 10 = 16$
So, the expression [6(1) + 5(2)\$} evaluates to ${16\$}.
Determining if the Point Satisfies the Inequality
Now that we have evaluated the expression, we can determine if the point {(1, 2)$}$ satisfies the inequality. We need to compare the result of the expression, ${16\$}, with the value on the right-hand side of the inequality, ${19\$}.
Since ${16\$} is less than ${19\$}, the point {(1, 2)$}$ satisfies the inequality.
Conclusion
In conclusion, the point {(1, 2)$}$ satisfies the inequality ${6x + 5y \ \textless \ 19?\$}. This means that the point lies within the region defined by the inequality. We can use this result to determine if other points satisfy the inequality by substituting their values into the inequality and evaluating the result.
Graphical Representation
To visualize the inequality, we can graph the line ${6x + 5y = 19\$} on a coordinate plane. The points that satisfy the inequality will lie below this line.
Graphing the Line
To graph the line ${6x + 5y = 19\$}, we can use the slope-intercept form of a linear equation, {y = mx + b$}$, where {m$}$ is the slope and {b$}$ is the y-intercept.
First, we need to isolate {y$}$ in the equation ${6x + 5y = 19\$}. We can do this by subtracting ${6x\$} from both sides of the equation and then dividing both sides by ${5\$}.
${$6x + 5y = 19$ [y = -\frac{6}{5}x + \frac{19}{5}$
Now that we have the equation in slope-intercept form, we can graph the line on a coordinate plane.
Graphing the Inequality
To graph the inequality [6x + 5y \ \textless \ 19?\$}, we can use the same line as before, {y = -\frac{6}{5}x + \frac{19}{5}$}$. However, we need to shade the region below the line to indicate that the points that satisfy the inequality lie below the line.
Shading the Region
To shade the region below the line, we can use a ruler or a straightedge to draw a line below the line {y = -\frac{6}{5}x + \frac{19}{5}$}$. This line should be parallel to the x-axis and should pass through the point {(0, 0)$}$.
Conclusion
In conclusion, the point {(1, 2)$}$ satisfies the inequality ${6x + 5y \ \textless \ 19?\$}. We can use the graphical representation of the inequality to visualize the region defined by the inequality and to determine if other points satisfy the inequality.
Final Answer
Introduction
In our previous article, we discussed whether a given point satisfies a specific inequality. We used the point {(1, 2)$}$ and the inequality ${6x + 5y \ \textless \ 19?\$} to determine if the point lies within the region defined by the inequality. In this article, we will answer some frequently asked questions related to the topic.
Q: What is a linear inequality?
A: A linear inequality is an inequality that can be represented graphically as a line on a coordinate plane. It is a statement that two expressions are not equal, but one is either greater than or less than the other.
Q: How do I graph a linear inequality?
A: To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality. If the inequality is of the form {ax + by \ \textless \ c?$}$, you will shade the region below the line. If the inequality is of the form {ax + by \ \textgreater \ c?$}$, you will shade the region above the line.
Q: How do I determine if a point satisfies a linear inequality?
A: To determine if a point satisfies a linear inequality, you need to substitute the values of the point into the inequality and evaluate the result. If the result is true, then the point satisfies the inequality.
Q: What is the difference between a linear inequality and a quadratic inequality?
A: A linear inequality is an inequality that can be represented graphically as a line on a coordinate plane. A quadratic inequality, on the other hand, is an inequality that can be represented graphically as a parabola on a coordinate plane.
Q: How do I solve a linear inequality?
A: To solve a linear inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.
Q: Can I use a calculator to solve a linear inequality?
A: Yes, you can use a calculator to solve a linear inequality. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct operations.
Q: What are some common mistakes to avoid when solving linear inequalities?
A: Some common mistakes to avoid when solving linear inequalities include:
- Not isolating the variable on one side of the inequality
- Not using the correct operations (e.g. adding or subtracting the same value to both sides of the inequality)
- Not checking the solution to make sure it satisfies the original inequality
- Not considering the direction of the inequality (e.g. {\textless $}$ vs. {\textgreater $}$)
Conclusion
In conclusion, solving linear inequalities requires a clear understanding of the concept of linear inequalities and the steps involved in solving them. By following the steps outlined in this article, you can confidently solve linear inequalities and determine if a point satisfies a linear inequality.
Final Answer
The final answer is: A. Yes
Additional Resources
For more information on linear inequalities, you can refer to the following resources:
- Khan Academy: Linear Inequalities
- Mathway: Linear Inequalities
- Wolfram Alpha: Linear Inequalities
Practice Problems
Try the following practice problems to test your understanding of linear inequalities:
- Does the point {(3, 4)$}$ satisfy the inequality ${2x + 3y \ \textless \ 12?\$}?
- Graph the inequality {x - 2y \ \textgreater \ 3?$}$ and determine if the point {(5, 2)$}$ satisfies the inequality.
- Solve the inequality ${4x - 2y \ \textless \ 10?\$} and determine if the point {(2, 3)$}$ satisfies the inequality.
Answer Key
- A. Yes
- A. Yes
- A. No