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)$}$

Introduction

Inequalities are mathematical expressions that compare two values, often with a variable or variables involved. Solving inequalities can be a challenging task, especially when we need to find the values of multiple variables that satisfy multiple conditions. In this article, we will explore how to solve a system of inequalities, specifically the ordered pair that makes both inequalities true.

Understanding the Inequalities

The given inequalities are:

1. $y < 3x - 1$
2. $y \geq -x + 4$

These inequalities represent two different conditions that the ordered pair $(x, y)$ must satisfy. The first inequality states that $y$ is less than $3x - 1$, while the second inequality states that $y$ is greater than or equal to $-x + 4$.

Graphing the Inequalities

To visualize the inequalities, we can graph them on a coordinate plane. The first inequality, $y < 3x - 1$, can be graphed as a line with a slope of 3 and a y-intercept of -1. The second inequality, $y \geq -x + 4$, can be graphed as a line with a slope of -1 and a y-intercept of 4.

**Graphing the Inequalities**

The first inequality, y < 3x - 1, is graphed as a line with a slope of 3 and a y-intercept of -1.
The second inequality, $y \geq -x + 4$, is graphed as a line with a slope of -1 and a y-intercept of 4.

Finding the Intersection

To find the ordered pair that satisfies both inequalities, we need to find the intersection of the two graphs. The intersection point represents the values of $x$ and $y$ that satisfy both conditions.

**Finding the Intersection**

To find the intersection point, we need to set the two equations equal to each other: $3x - 1 = -x + 4$.
Solving for $x$, we get $4x = 5$, which gives us $x = \frac{5}{4}$.
Substituting $x = \frac{5}{4}$ into one of the original equations, we can solve for $y$.

Solving for $y$

Substituting $x = \frac{5}{4}$ into the first equation, $y < 3x - 1$, we get:

$y < 3(\frac{5}{4}) - 1$

Simplifying the equation, we get:

$y < \frac{15}{4} - 1$

$y < \frac{11}{4}$

Finding the Ordered Pair

Now that we have the values of $x$ and $y$, we can find the ordered pair that satisfies both inequalities.

**Finding the Ordered Pair**

The ordered pair that satisfies both inequalities is $(\frac{5}{4}, \frac{11}{4})$.
However, this ordered pair is not among the answer choices.
We need to check the answer choices to see if any of them satisfy both inequalities.

Checking the Answer Choices

Let's check each of the answer choices to see if any of them satisfy both inequalities.

Answer Choice A: $(4, 0)$

Substituting $x = 4$ and $y = 0$ into the first inequality, $y < 3x - 1$, we get:

$0 < 3(4) - 1$

$0 < 12 - 1$

$0 < 11$

This is true, so the first inequality is satisfied.

Substituting $x = 4$ and $y = 0$ into the second inequality, $y \geq -x + 4$, we get:

$0 \geq -4 + 4$

$0 \geq 0$

This is true, so the second inequality is also satisfied.

Answer Choice B: $(1, 2)$

Substituting $x = 1$ and $y = 2$ into the first inequality, $y < 3x - 1$, we get:

$2 < 3(1) - 1$

$2 < 2$

This is false, so the first inequality is not satisfied.

Substituting $x = 1$ and $y = 2$ into the second inequality, $y \geq -x + 4$, we get:

$2 \geq -1 + 4$

$2 \geq 3$

This is false, so the second inequality is not satisfied.

Answer Choice C: $(0, 4)$

Substituting $x = 0$ and $y = 4$ into the first inequality, $y < 3x - 1$, we get:

$4 < 3(0) - 1$

$4 < -1$

This is false, so the first inequality is not satisfied.

Substituting $x = 0$ and $y = 4$ into the second inequality, $y \geq -x + 4$, we get:

$4 \geq -0 + 4$

$4 \geq 4$

This is true, so the second inequality is satisfied.

Conclusion

In conclusion, the ordered pair that satisfies both inequalities is $(4, 0)$. This is the only answer choice that satisfies both inequalities.

Final Answer

The final answer is: $\boxed{(4, 0)}$

Introduction

In our previous article, we explored how to solve a system of inequalities and find the ordered pair that satisfies both conditions. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is the difference between a linear inequality and a nonlinear 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 nonlinear inequality, on the other hand, is an inequality that cannot be written in this form.

Q: How do I graph a linear inequality on a coordinate plane?

A: To graph a linear inequality on a coordinate plane, you need to graph the corresponding equation and then shade the region that satisfies the inequality. If the inequality is of the form $y < mx + b$, you will shade the region below the line. If the inequality is of the form $y > mx + b$, you will shade the region above the line.

Q: How do I find the intersection of two linear inequalities?

A: To find the intersection of two linear inequalities, you need to solve the system of equations formed by the two inequalities. You can do this by setting the two equations equal to each other and solving for the variable.

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 $\leq$ or $\geq$.

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

A: To solve a system of linear inequalities, you need to find the values of the variables that satisfy all the inequalities. You can do this by graphing the inequalities on a coordinate plane and finding the intersection of the graphs.

Q: What is the importance of solving inequalities in real-life situations?

A: Solving inequalities is an important skill in many real-life situations, such as finance, economics, and engineering. For example, a company may want to know the maximum amount of money it can spend on a project, given a certain budget constraint. Solving inequalities can help the company determine the maximum amount of money it can spend.

Q: How do I check if an ordered pair satisfies a linear inequality?

A: To check if an ordered pair satisfies a linear inequality, you need to substitute the values of the variables into the inequality and check if the inequality is true.

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 quadratic inequality?

A: To solve a quadratic inequality, you need to find the values of the variable that satisfy the inequality. You can do this by factoring the quadratic expression and then using the sign of the expression to determine the values of the variable that satisfy the inequality.

Conclusion

In conclusion, solving inequalities is an important skill in mathematics and has many real-life applications. By understanding the concepts of linear and nonlinear inequalities, graphing inequalities, and solving systems of inequalities, you can develop a strong foundation in solving inequalities.

Final Answer

The final answer is: There is no final numerical answer to this article. The article is a Q&A guide to solving inequalities.