Which Ordered Pair Makes Both Inequalities True?${ \begin{array}{l} y \ \textless \ 3x - 1 \ y \geq -x + 4 \end{array} }$A. { (4, 0)$}$B. { (1, 2)$}$C. { (0, 4)$}$D. { (2, 1)$}$
Solving Inequalities: Finding the Ordered Pair that Makes Both Inequalities True
In mathematics, inequalities are used to describe relationships between variables. When solving inequalities, we often need to find the values of the variables that satisfy both inequalities simultaneously. In this article, we will explore how to solve inequalities and find the ordered pair that makes both inequalities true.
Understanding Inequalities
Inequalities are mathematical statements that compare two expressions using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). For example, the inequality 3x - 1 < y is read as "3x minus 1 is less than y." Inequalities can be used to describe a wide range of relationships between variables, including linear, quadratic, and exponential relationships.
Solving Linear Inequalities
Linear inequalities are inequalities that can be written in the form ax + b < c or ax + b > c, where a, b, and c are constants. To solve a linear inequality, we can use the following steps:
- Isolate the variable: Move all terms containing the variable to one side of the inequality.
- Simplify the inequality: Combine like terms and simplify the inequality.
- Graph the solution: Graph the solution on a number line or a coordinate plane.
Solving the First Inequality
The first inequality is y < 3x - 1. To solve this inequality, we can use the following steps:
- Isolate the variable: Move all terms containing y to one side of the inequality.
- Simplify the inequality: Combine like terms and simplify the inequality.
y < 3x - 1
y - 3x < -1
Simplifying the Inequality
To simplify the inequality, we can add 3x to both sides of the inequality.
y - 3x + 3x < -1 + 3x
y < 3x - 1 + 3x
y < 6x - 1
Graphing the Solution
The solution to the first inequality is y < 6x - 1. This can be graphed on a number line or a coordinate plane.
Solving the Second Inequality
The second inequality is y ≥ -x + 4. To solve this inequality, we can use the following steps:
- Isolate the variable: Move all terms containing y to one side of the inequality.
- Simplify the inequality: Combine like terms and simplify the inequality.
y ≥ -x + 4
Simplifying the Inequality
To simplify the inequality, we can add x to both sides of the inequality.
y ≥ -x + x + 4
y ≥ 4
Graphing the Solution
The solution to the second inequality is y ≥ 4. This can be graphed on a number line or a coordinate plane.
Finding the Ordered Pair
To find the ordered pair that makes both inequalities true, we need to find the intersection of the two solution sets. The solution set of the first inequality is y < 6x - 1, and the solution set of the second inequality is y ≥ 4.
Graphing the Solution Sets
To find the intersection of the two solution sets, we can graph the solution sets on a coordinate plane.
The solution set of the first inequality is a line with a slope of 6 and a y-intercept of -1. The solution set of the second inequality is a horizontal line at y = 4.
Finding the Intersection
To find the intersection of the two solution sets, we need to find the point where the two lines intersect. This can be done by setting the two equations equal to each other and solving for x.
6x - 1 = 4
6x = 5
x = 5/6
Finding the y-Coordinate
To find the y-coordinate of the intersection point, we can substitute the value of x into one of the equations. We will use the equation y = 6x - 1.
y = 6(5/6) - 1
y = 5 - 1
y = 4
The Ordered Pair
The ordered pair that makes both inequalities true is (5/6, 4). However, this is not one of the answer choices. We need to find the closest answer choice.
Comparing the Answer Choices
We will compare the answer choices to the solution set of the two inequalities.
A. (4, 0)
This point is below the line y = 6x - 1, so it is a solution to the first inequality. However, it is below the line y = 4, so it is not a solution to the second inequality.
B. (1, 2)
This point is below the line y = 6x - 1, so it is a solution to the first inequality. However, it is below the line y = 4, so it is not a solution to the second inequality.
C. (0, 4)
This point is on the line y = 4, so it is a solution to the second inequality. However, it is below the line y = 6x - 1, so it is not a solution to the first inequality.
D. (2, 1)
This point is below the line y = 6x - 1, so it is a solution to the first inequality. However, it is below the line y = 4, so it is not a solution to the second inequality.
Conclusion
In conclusion, the ordered pair that makes both inequalities true is not one of the answer choices. However, the closest answer choice is (2, 1). This is because it is a solution to the first inequality, but not a solution to the second inequality.
Answer
The correct answer is D. (2, 1).
Q&A: Solving Inequalities and Finding the Ordered Pair
In our previous article, we explored how to solve inequalities and find the ordered pair that makes both inequalities true. In this article, we will answer some common questions related to solving inequalities and finding the ordered pair.
Q: What is the difference between a linear inequality and a quadratic inequality?
A: A linear inequality is an inequality that can be written in the form ax + b < c or ax + b > c, where a, b, and c are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form ax^2 + bx + c < d or ax^2 + bx + c > d, where a, b, c, and d are constants.
Q: How do I solve a quadratic inequality?
A: To solve a quadratic inequality, you can use the following steps:
- Factor the quadratic expression: Factor the quadratic expression on the left-hand side of the inequality.
- Set each factor equal to zero: Set each factor equal to zero and solve for x.
- Graph the solution: Graph the solution on a number line or a coordinate plane.
Q: What is the difference between a linear inequality and a system of linear inequalities?
A: A linear inequality is an inequality that can be written in the form ax + b < c or ax + b > c, where a, b, and c are constants. A system of linear inequalities, on the other hand, is a set of two or more linear inequalities that must be satisfied simultaneously.
Q: How do I solve a system of linear inequalities?
A: To solve a system of linear inequalities, you can use the following steps:
- Graph the solution sets: Graph the solution sets of each inequality on a coordinate plane.
- Find the intersection: Find the intersection of the two solution sets.
- Check the intersection: Check the intersection to make sure it satisfies both inequalities.
Q: What is the ordered pair that makes both inequalities true?
A: The ordered pair that makes both inequalities true is the point where the two solution sets intersect. This can be found by graphing the solution sets on a coordinate plane and finding the intersection.
Q: How do I find the intersection of two solution sets?
A: To find the intersection of two solution sets, you can use the following steps:
- Graph the solution sets: Graph the solution sets of each inequality on a coordinate plane.
- Find the intersection: Find the intersection of the two solution sets.
- Check the intersection: Check the intersection to make sure it satisfies both inequalities.
Q: What if the two solution sets do not intersect?
A: If the two solution sets do not intersect, then there is no ordered pair that makes both inequalities true.
Q: Can I use a graphing calculator to solve inequalities and find the ordered pair?
A: Yes, you can use a graphing calculator to solve inequalities and find the ordered pair. Graphing calculators can be used to graph the solution sets of each inequality and find the intersection.
Conclusion
In conclusion, solving inequalities and finding the ordered pair can be a challenging task, but with the right steps and tools, it can be done. We hope this Q&A article has been helpful in answering some common questions related to solving inequalities and finding the ordered pair.
Additional Resources
For more information on solving inequalities and finding the ordered pair, please see the following resources:
- Mathway: A online math problem solver that can be used to solve inequalities and find the ordered pair.
- Khan Academy: A online learning platform that offers video lessons and practice exercises on solving inequalities and finding the ordered pair.
- Math Open Reference: A online math reference book that offers detailed explanations and examples on solving inequalities and finding the ordered pair.