The Inequality $x + 2y \geq 3$ Is Satisfied By Point $(1, 1)$. Type true Or false In The Space Provided.

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Introduction

In mathematics, inequalities are used to describe the relationship between two or more variables. They are an essential part of mathematical modeling and problem-solving. In this article, we will discuss the inequality $x + 2y \geq 3$ and determine whether it is satisfied by the point $(1, 1)$.

Understanding the Inequality

The given inequality is $x + 2y \geq 3$. This is a linear inequality in two variables, $x$ and $y$. The inequality states that the sum of $x$ and $2y$ is greater than or equal to $3$. To determine whether the point $(1, 1)$ satisfies this inequality, we need to substitute the values of $x$ and $y$ into the inequality and check if it holds true.

Substituting the Values of $x$ and $y$

Let's substitute the values of $x$ and $y$ into the inequality $x + 2y \geq 3$. We have $x = 1$ and $y = 1$. Substituting these values into the inequality, we get:

$$1 + 2(1) \geq 3$$

Evaluating the Inequality

Now, let's evaluate the inequality. We have:

$$1 + 2(1) = 1 + 2 = 3$$

The inequality becomes:

$$3 \geq 3$$

Conclusion

Since $3 \geq 3$ is a true statement, the point $(1, 1)$ satisfies the inequality $x + 2y \geq 3$. Therefore, the answer is true.

Understanding the Graphical Representation

To visualize the inequality $x + 2y \geq 3$, we can graph the corresponding equation $x + 2y = 3$. This equation represents a line in the coordinate plane. The inequality $x + 2y \geq 3$ represents all the points on or above this line.

Graphing the Line

To graph the line $x + 2y = 3$, we can use the slope-intercept form of a linear equation, which is $y = mx + b$. We can rewrite the equation $x + 2y = 3$ as $2y = -x + 3$, and then divide both sides by $2$ to get $y = -\frac{1}{2}x + \frac{3}{2}$. This is the slope-intercept form of the equation.

Finding the $y$-Intercept

The $y$-intercept of the line is the point where the line intersects the $y$-axis. To find the $y$-intercept, we can set $x = 0$ and solve for $y$. We have:

$$y = -\frac{1}{2}(0) + \frac{3}{2} = \frac{3}{2}$$

Finding the $x$-Intercept

The $x$-intercept of the line is the point where the line intersects the $x$-axis. To find the $x$-intercept, we can set $y = 0$ and solve for $x$. We have:

$$0 = -\frac{1}{2}x + \frac{3}{2}$$

Solving for $x$, we get:

$$x = 3$$

Graphing the Inequality

To graph the inequality $x + 2y \geq 3$, we can graph the corresponding equation $x + 2y = 3$ and shade the region on or above the line. This represents all the points that satisfy the inequality.

Conclusion

In conclusion, the point $(1, 1)$ satisfies the inequality $x + 2y \geq 3$. We can also graph the inequality by graphing the corresponding equation and shading the region on or above the line.

Final Answer

The final answer is true.

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Introduction

In our previous article, we discussed the inequality $x + 2y \geq 3$ and determined whether it is satisfied by the point $(1, 1)$. In this article, we will answer some frequently asked questions related to the inequality and the point.

Q&A

Q: What is the inequality $x + 2y \geq 3$?

A: The inequality $x + 2y \geq 3$ is a linear inequality in two variables, $x$ and $y$. It states that the sum of $x$ and $2y$ is greater than or equal to $3$.

Q: How do I determine whether a point satisfies the inequality?

A: To determine whether a point satisfies the inequality, you need to substitute the values of $x$ and $y$ into the inequality and check if it holds true.

Q: What is the corresponding equation of the inequality?

A: The corresponding equation of the inequality $x + 2y \geq 3$ is $x + 2y = 3$.

Q: How do I graph the inequality?

A: To graph the inequality, you can graph the corresponding equation $x + 2y = 3$ and shade the region on or above the line.

Q: What is the $y$-intercept of the line?

A: The $y$-intercept of the line is the point where the line intersects the $y$-axis. To find the $y$-intercept, you can set $x = 0$ and solve for $y$.

Q: What is the $x$-intercept of the line?

A: The $x$-intercept of the line is the point where the line intersects the $x$-axis. To find the $x$-intercept, you can set $y = 0$ and solve for $x$.

Q: How do I determine whether the point $(1, 1)$ satisfies the inequality?

A: To determine whether the point $(1, 1)$ satisfies the inequality, you need to substitute the values of $x$ and $y$ into the inequality and check if it holds true. We have $x = 1$ and $y = 1$. Substituting these values into the inequality, we get:

$$1 + 2(1) \geq 3$$

Evaluating the inequality, we get:

$$3 \geq 3$$

Since $3 \geq 3$ is a true statement, the point $(1, 1)$ satisfies the inequality.

Q: What is the final answer?

A: The final answer is true.

Conclusion

In conclusion, we have answered some frequently asked questions related to the inequality $x + 2y \geq 3$ and the point $(1, 1)$. We hope that this article has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer is true.

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• Linear Inequalities
• Graphing Linear Inequalities
• Slope-Intercept Form

Note: The above resources are provided for additional information and are not part of the main article.