Select The Correct Answer.Which Point Is A Solution To This System Of Inequalities?${ \begin{align} y & \leq \frac{1}{2} X - 3 \ y + 2x & \ \textgreater \ 6 \end{align} }$A. { (2, -3)$}$ B. { (7, -8)$}$ C.

Mar 13, 2025 by ADMIN 213 views

**Solving Systems of Inequalities: A Step-by-Step Guide**

Understanding Systems of Inequalities

A system of inequalities is a set of two or more inequalities that are related to each other. In this article, we will focus on solving systems of inequalities that involve linear inequalities. These inequalities are in the form of $y \leq mx + b$ or $y \geq mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

The Given System of Inequalities

The given system of inequalities is:

{ \begin{align*} y & \leq \frac{1}{2} x - 3 \\ y + 2x & \ \textgreater \ 6 \end{align*} \}

To solve this system, we need to find the solution that satisfies both inequalities.

Step 1: Rewrite the Second Inequality

The second inequality can be rewritten as:

$y \ \textgreater \ -2x + 6$

This is because we can isolate $y$ by subtracting $2x$ from both sides of the inequality.

Step 2: Graph the First Inequality

The first inequality is $y \leq \frac{1}{2} x - 3$ . To graph this inequality, we need to find the y-intercept, which is $-3$ . We also need to find the x-intercept, which is $-6$ . The graph of this inequality is a line with a slope of $\frac{1}{2}$ and a y-intercept of $-3$ .

Step 3: Graph the Second Inequality

The second inequality is $y \ \textgreater \ -2x + 6$ . To graph this inequality, we need to find the y-intercept, which is $6$ . We also need to find the x-intercept, which is $-3$ . The graph of this inequality is a line with a slope of $-2$ and a y-intercept of $6$ .

Step 4: Find the Solution

To find the solution, we need to find the region that satisfies both inequalities. The region that satisfies the first inequality is the region below the line $y = \frac{1}{2} x - 3$ . The region that satisfies the second inequality is the region above the line $y = -2x + 6$ .

Step 5: Check the Answer Choices

We are given three answer choices: $(2, -3)$ , $(7, -8)$ , and $(4, -5)$ . We need to check which of these answer choices satisfies both inequalities.

Checking Answer Choice A

Answer choice A is $(2, -3)$ . We can plug this point into both inequalities to check if it satisfies both inequalities.

For the first inequality, we have:

$-3 \leq \frac{1}{2} (2) - 3$

Simplifying, we get:

$-3 \leq -2$

This is true, so the point $(2, -3)$ satisfies the first inequality.

For the second inequality, we have:

$-3 \ \textgreater \ -2(2) + 6$

Simplifying, we get:

$-3 \ \textgreater \ 2$

This is true, so the point $(2, -3)$ satisfies the second inequality.

Checking Answer Choice B

Answer choice B is $(7, -8)$ . We can plug this point into both inequalities to check if it satisfies both inequalities.

For the first inequality, we have:

$-8 \leq \frac{1}{2} (7) - 3$

Simplifying, we get:

$-8 \leq -2$

This is true, so the point $(7, -8)$ satisfies the first inequality.

For the second inequality, we have:

$-8 \ \textgreater \ -2(7) + 6$

Simplifying, we get:

$-8 \ \textgreater \ -8$

This is false, so the point $(7, -8)$ does not satisfy the second inequality.

Conclusion

Based on our analysis, we can conclude that the correct answer is:

A. $(2, -3)$

This point satisfies both inequalities and is therefore the solution to the system of inequalities.

Discussion

Systems of inequalities are an important topic in mathematics, and solving them requires a deep understanding of linear inequalities and graphing. In this article, we have seen how to solve a system of inequalities by graphing the individual inequalities and finding the region that satisfies both inequalities. We have also seen how to check answer choices by plugging them into both inequalities.

Tips and Tricks

When solving systems of inequalities, it is often helpful to graph the individual inequalities on a coordinate plane.
To find the solution, look for the region that satisfies both inequalities.
When checking answer choices, make sure to plug them into both inequalities to ensure that they satisfy both inequalities.

Practice Problems

Solve the following system of inequalities:

{ \begin{align*} y & \leq 2x - 1 \\ y + 3x & \ \textgreater \ 5 \end{align*} \}

Solve the following system of inequalities:

{ \begin{align*} y & \geq x + 2 \\ y - 2x & \ \textless \ 3 \end{align*} \}

References

[1] "Systems of Inequalities" by Math Open Reference
[2] "Solving Systems of Inequalities" by Khan Academy

Q: What is a system of inequalities?

A: A system of inequalities is a set of two or more inequalities that are related to each other. In this article, we will focus on solving systems of inequalities that involve linear inequalities.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to find the solution that satisfies both inequalities. This can be done by graphing the individual inequalities on a coordinate plane and finding the region that satisfies both inequalities.

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

A: A system of equations is a set of two or more equations that are related to each other. A system of inequalities, on the other hand, is a set of two or more inequalities that are related to each other. In a system of equations, the solution is a specific point that satisfies all the equations, while in a system of inequalities, the solution is a region that satisfies all the inequalities.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to find the y-intercept and the x-intercept of the inequality. The y-intercept is the point where the inequality crosses the y-axis, and the x-intercept is the point where the inequality crosses the x-axis. You can then use these points to draw the graph of the inequality.

Q: What is the solution to a system of inequalities?

A: The solution to a system of inequalities is the region that satisfies all the inequalities. This region is the area where all the inequalities are true.

Q: How do I check if a point is a solution to a system of inequalities?

A: To check if a point is a solution to a system of inequalities, you need to plug the point into both inequalities and check if it satisfies both inequalities.

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 is an inequality that is written with a non-strict symbol, such as $\leq$ or $\geq$ . In a strict inequality, the solution is a single point, while in a non-strict inequality, the solution is a region.

Q: How do I solve a system of inequalities with two variables?

A: To solve a system of inequalities with two variables, you need to find the solution that satisfies both inequalities. This can be done by graphing the individual inequalities on a coordinate plane and finding the region that satisfies both inequalities.

Q: What is the importance of solving systems of inequalities?

A: Solving systems of inequalities is an important topic in mathematics because it helps us to understand how to solve real-world problems that involve multiple constraints. In many real-world problems, we need to find the solution that satisfies multiple constraints, and solving systems of inequalities is a key tool for doing this.

Q: How do I apply systems of inequalities to real-world problems?

A: To apply systems of inequalities to real-world problems, you need to identify the constraints of the problem and write them as inequalities. You can then use the techniques we have discussed to solve the system of inequalities and find the solution.

Q: What are some common applications of systems of inequalities?

A: Some common applications of systems of inequalities include:

Finance: Systems of inequalities can be used to model financial problems, such as finding the minimum or maximum value of an investment.
Engineering: Systems of inequalities can be used to model engineering problems, such as finding the optimal design of a system.
Economics: Systems of inequalities can be used to model economic problems, such as finding the optimal price of a good.

Q: How do I practice solving systems of inequalities?

A: To practice solving systems of inequalities, you can try the following:

Practice problems: Try solving systems of inequalities with different numbers of variables and different types of inequalities.
Real-world problems: Try applying systems of inequalities to real-world problems, such as finance, engineering, or economics.
Online resources: Try using online resources, such as Khan Academy or Math Open Reference, to practice solving systems of inequalities.

Q: What are some common mistakes to avoid when solving systems of inequalities?

A: Some common mistakes to avoid when solving systems of inequalities include:

Not graphing the inequalities correctly: Make sure to graph the inequalities correctly, including the y-intercept and the x-intercept.
Not checking the solution: Make sure to check the solution to ensure that it satisfies both inequalities.
Not considering the constraints: Make sure to consider the constraints of the problem and write them as inequalities.

Q: How do I know if I have solved the system of inequalities correctly?

A: To know if you have solved the system of inequalities correctly, you need to check the solution to ensure that it satisfies both inequalities. You can also use online resources, such as Khan Academy or Math Open Reference, to check your solution.