Which Point Is A Solution To $y \leq 4x + 5$?A. ( − 6 , 4 (-6, 4 ( − 6 , 4 ] B. ( − 4 , 0 (-4, 0 ( − 4 , 0 ] C. ( 0 , 10 (0, 10 ( 0 , 10 ] D. ( 0 , − 2 (0, -2 ( 0 , − 2 ]

Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear inequalities in the form of $y \leq mx + b$, where $m$ is the slope and $b$ is the y-intercept. We will use a step-by-step approach to solve the given inequality and find the solution point.

Understanding Linear Inequalities

A linear inequality is an inequality that can be written in the form of $y \leq mx + b$, where $m$ is the slope and $b$ is the y-intercept. The inequality can be either less than or equal to ($\leq$) or greater than or equal to ($\geq$). In this article, we will focus on solving the inequality $y \leq 4x + 5$.

Graphing the Inequality

To solve the inequality $y \leq 4x + 5$, we need to graph the related equation $y = 4x + 5$. The graph of the equation is a straight line with a slope of 4 and a y-intercept of 5.

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

y = 4*x + 5

plt.plot(x, y)
plt.title('Graph of y = 4x + 5')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Finding the Solution Point

To find the solution point, we need to find the point that satisfies the inequality $y \leq 4x + 5$. We can do this by substituting the x and y values of each point into the inequality and checking if it is true.

Let's substitute the x and y values of each point into the inequality:

• A. $(-6, 4)$: $4 \leq 4(-6) + 5$
• B. $(-4, 0)$: $0 \leq 4(-4) + 5$
• C. $(0, 10)$: $10 \leq 4(0) + 5$
• D. $(0, -2)$: $-2 \leq 4(0) + 5$

Solving the Inequality

Now, let's solve the inequality for each point:

• A. $4 \leq -24 + 5$
• B. $0 \leq -16 + 5$
• C. $10 \leq 0 + 5$
• D. $-2 \leq 0 + 5$

Simplifying the inequality for each point, we get:

• A. $4 \leq -19$
• B. $0 \leq -11$
• C. $10 \leq 5$
• D. $-2 \leq 5$

Since the inequality is not true for points A, B, and C, the only point that satisfies the inequality is point D.

Conclusion

In conclusion, the solution to the inequality $y \leq 4x + 5$ is point D, which is $(0, -2)$. This point satisfies the inequality, and it is the only point that does so.

Final Answer

The final answer is:

• D. $(0, -2)$
  Solving Linear Inequalities: A Q&A Guide =====================================================

Introduction

In our previous article, we discussed how to solve linear inequalities in the form of $y \leq mx + b$. We used a step-by-step approach to solve the inequality $y \leq 4x + 5$ and found the solution point. In this article, we will provide a Q&A guide to help students understand and solve linear inequalities.

Q&A Guide

Q1: What is a linear inequality?

A linear inequality is an inequality that can be written in the form of $y \leq mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q2: How do I graph a linear inequality?

To graph a linear inequality, you need to graph the related equation $y = mx + b$. The graph of the equation is a straight line with a slope of $m$ and a y-intercept of $b$.

Q3: How do I find the solution point of a linear inequality?

To find the solution point, you need to substitute the x and y values of each point into the inequality and check if it is true. You can use the following steps:

• Substitute the x and y values of each point into the inequality.
• Simplify the inequality and check if it is true.
• If the inequality is true, then the point is a solution point.

Q4: What is the difference between a linear inequality and a linear equation?

A linear equation is an equation that can be written in the form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. A linear inequality is an inequality that can be written in the form of $y \leq mx + b$ or $y \geq mx + b$.

Q5: How do I solve a linear inequality with a negative slope?

To solve a linear inequality with a negative slope, you need to follow the same steps as solving a linear inequality with a positive slope. The only difference is that the inequality will be in the form of $y \geq mx + b$ or $y \leq mx + b$, where $m$ is a negative number.

Q6: Can I use a graphing calculator to solve a linear inequality?

Yes, you can use a graphing calculator to solve a linear inequality. You can graph the related equation $y = mx + b$ and then use the calculator to find the solution points.

Q7: How do I check if a point is a solution point of a linear inequality?

To check if a point is a solution point, you need to substitute the x and y values of the point into the inequality and check if it is true. If the inequality is true, then the point is a solution point.

Q8: Can I have multiple solution points for a linear inequality?

Yes, you can have multiple solution points for a linear inequality. This occurs when the inequality is in the form of $y \leq mx + b$ or $y \geq mx + b$, and there are multiple points that satisfy the inequality.

Q9: How do I write a linear inequality in the form of $y \leq mx + b$ or $y \geq mx + b$?

To write a linear inequality in the form of $y \leq mx + b$ or $y \geq mx + b$, you need to follow these steps:

• Write the inequality in the form of $y = mx + b$.
• Replace the $=$ sign with a $\leq$ or $\geq$ sign.
• Simplify the inequality.

Q10: Can I use a linear inequality to model real-world problems?

Yes, you can use a linear inequality to model real-world problems. Linear inequalities can be used to model situations where there are constraints or limitations.

Conclusion

In conclusion, linear inequalities are an important concept in mathematics, and solving them is a crucial skill for students to master. By following the steps outlined in this Q&A guide, students can learn how to solve linear inequalities and apply them to real-world problems.

Final Answer

The final answer is:

• D. $(0, -2)$