Which Ordered Pair Makes Both Inequalities True?${ \begin{align*} y & \leq -x + 1 \ y & \ \textgreater \ X \end{align*} }$

Introduction

In mathematics, inequalities are used to describe relationships between variables. When we have two inequalities, we need to find the values of the variables that satisfy both inequalities simultaneously. In this article, we will explore how to find the ordered pair that makes both inequalities true.

Understanding the Inequalities

The two inequalities given are:

1. $y \leq -x + 1$
2. $y > x$

To find the ordered pair that satisfies both inequalities, we need to understand the meaning of each inequality.

Inequality 1: $y \leq -x + 1$

This inequality states that $y$ is less than or equal to $-x + 1$. This means that for any value of $x$, the corresponding value of $y$ must be less than or equal to $-x + 1$.

Inequality 2: $y > x$

This inequality states that $y$ is greater than $x$. This means that for any value of $x$, the corresponding value of $y$ must be greater than $x$.

Finding the Ordered Pair

To find the ordered pair that satisfies both inequalities, we need to find the values of $x$ and $y$ that satisfy both inequalities simultaneously.

Graphing the Inequalities

One way to visualize the inequalities is to graph them on a coordinate plane. The graph of the first inequality is a line with a slope of $-1$ and a $y$-intercept of $1$. The graph of the second inequality is a line with a slope of $1$ and a $y$-intercept of $0$.

import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(-10, 10, 400)

y1 = -x + 1

y2 = x

plt.plot(x, y1, label='y <= -x + 1')
plt.plot(x, y2, label='y > x')

plt.title('Graph of the Inequalities')
plt.xlabel('x')
plt.ylabel('y')

plt.legend()

plt.show()

Finding the Intersection Point

The ordered pair that satisfies both inequalities must lie on both lines. To find this point, we need to find the intersection point of the two lines.

To find the intersection point, we can set the two equations equal to each other and solve for $x$.

$-x + 1 = x$

Solving for $x$, we get:

$2x = 1$

$x = \frac{1}{2}$

Now that we have the value of $x$, we can substitute it into one of the equations to find the value of $y$.

$y = -x + 1$

$y = -\frac{1}{2} + 1$

$y = \frac{1}{2}$

Therefore, the ordered pair that satisfies both inequalities is $\left(\frac{1}{2}, \frac{1}{2}\right)$.

Conclusion

In this article, we explored how to find the ordered pair that makes both inequalities true. We graphed the inequalities on a coordinate plane and found the intersection point of the two lines. The ordered pair that satisfies both inequalities is $\left(\frac{1}{2}, \frac{1}{2}\right)$.

Frequently Asked Questions

Q: What is the meaning of the inequality $y \leq -x + 1$?

A: This inequality states that $y$ is less than or equal to $-x + 1$. This means that for any value of $x$, the corresponding value of $y$ must be less than or equal to $-x + 1$.

Q: What is the meaning of the inequality $y > x$?

A: This inequality states that $y$ is greater than $x$. This means that for any value of $x$, the corresponding value of $y$ must be greater than $x$.

Q: How do I graph the inequalities on a coordinate plane?

A: To graph the inequalities, you can use a graphing calculator or a computer program such as Python. You can also use a coordinate plane and plot the points that satisfy the inequalities.

Q: How do I find the intersection point of the two lines?

A: To find the intersection point, you can set the two equations equal to each other and solve for $x$. Then, you can substitute the value of $x$ into one of the equations to find the value of $y$.

References

• [1] "Inequalities" by Khan Academy
• [2] "Graphing Inequalities" by Math Open Reference
• [3] "Finding the Intersection Point of Two Lines" by Purplemath