Which Ordered Pair Makes Both Inequalities True?${ \begin{array}{l} y \ \textgreater \ -2x + 3 \ y \leq X - 2 \end{array} }$A. { (0, 0)$}$ B. { (0, -1)$}$ C. { (1, 1)$}$ D. { (3, 0)$}$

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 conditions. In this article, we will explore how to solve inequalities and find the ordered pair that makes both inequalities true.

Understanding Inequalities

An inequality is a statement that compares two expressions, indicating whether one is greater than, less than, or equal to the other. Inequalities can be written in various forms, including:

• Greater than (>): $y > -2x + 3$
• Less than or equal to (≤): $y \leq x - 2$

Solving the First Inequality

The first inequality is $y > -2x + 3$. To solve this inequality, we need to isolate the variable $y$. We can do this by adding $2x$ to both sides of the inequality:

$$y > -2x + 3$$

$$y + 2x > 3$$

Solving the Second Inequality

The second inequality is $y \leq x - 2$. To solve this inequality, we need to isolate the variable $y$. We can do this by subtracting $x$ from both sides of the inequality:

$$y \leq x - 2$$

$$y - x \leq -2$$

Finding the Ordered Pair

Now that we have solved both inequalities, we need to find the ordered pair that satisfies both conditions. We can do this by finding the intersection of the two solution sets.

Graphing the Inequalities

To visualize the solution sets, we can graph the inequalities on a coordinate plane. The first inequality, $y > -2x + 3$, can be graphed as a line with a slope of $-2$ and a y-intercept of $3$. The second inequality, $y \leq x - 2$, can be graphed as a line with a slope of $1$ and a y-intercept of $-2$.

Finding the Intersection

The intersection of the two solution sets is the region where both inequalities are true. To find the intersection, we need to find the point where the two lines intersect.

Analyzing the Options

We are given four options: $(0, 0)$, $(0, -1)$, $(1, 1)$, and $(3, 0)$. We need to analyze each option to see if it satisfies both inequalities.

Option A: $(0, 0)$

Let's substitute $x = 0$ and $y = 0$ into both inequalities:

• First inequality: $0 > -2(0) + 3$ is false
• Second inequality: $0 \leq 0 - 2$ is true

Since the first inequality is false, option A is not a solution.

Option B: $(0, -1)$

Let's substitute $x = 0$ and $y = -1$ into both inequalities:

• First inequality: $-1 > -2(0) + 3$ is false
• Second inequality: $-1 \leq 0 - 2$ is true

Since the first inequality is false, option B is not a solution.

Option C: $(1, 1)$

Let's substitute $x = 1$ and $y = 1$ into both inequalities:

• First inequality: $1 > -2(1) + 3$ is true
• Second inequality: $1 \leq 1 - 2$ is true

Since both inequalities are true, option C is a solution.

Option D: $(3, 0)$

Let's substitute $x = 3$ and $y = 0$ into both inequalities:

• First inequality: $0 > -2(3) + 3$ is false
• Second inequality: $0 \leq 3 - 2$ is true

Since the first inequality is false, option D is not a solution.

Conclusion

In conclusion, the ordered pair that makes both inequalities true is $(1, 1)$. This is the only option that satisfies both conditions.

Final Answer

The final answer is $\boxed{(1, 1)}$.

Introduction

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 frequently asked questions about solving inequalities.

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 $ax + by < c$, where $a$, $b$, and $c$ are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c < 0$, where $a$, $b$, and $c$ are constants.

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: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict symbol, such as $>$ or $<$. A non-strict inequality, on the other hand, is an inequality that is written with a non-strict symbol, such as $\geq$ or $\leq$.

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 strict, you should shade the region on one side of the line. If the inequality is non-strict, you should shade the region on both sides of the line.

Q: What is the intersection of two solution sets?

A: The intersection of two solution sets is the region where both solution sets overlap. To find the intersection, you need to find the point where the two solution sets intersect.

Q: How do I find the intersection of two solution sets?

A: To find the intersection of two solution sets, you need to solve the system of equations that corresponds to the two solution sets. You can do this by using substitution or elimination methods.

Q: What is the difference between a system of linear inequalities and a system of linear equations?

A: A system of linear inequalities is a set of linear inequalities that are all true at the same time. A system of linear equations, on the other hand, is a set of linear equations that are all true at the same time.

Q: How do I solve a system of linear inequalities?

A: To solve a system of linear inequalities, you need to find the solution set that satisfies all the inequalities in the system. You can do this by graphing the inequalities and finding the intersection of the solution sets.

Q: What is the final answer to the problem of finding the ordered pair that makes both inequalities true?

A: The final answer to the problem of finding the ordered pair that makes both inequalities true is $(1, 1)$.

Conclusion

In conclusion, solving inequalities is an important topic in mathematics that requires a deep understanding of linear and quadratic equations, as well as graphing and systems of equations. By following the steps outlined in this article, you should be able to solve inequalities and find the ordered pair that makes both inequalities true.

Final Answer

The final answer is $\boxed{(1, 1)}$.