MP3 Essential Question: Although Jess Got The Correct Answer, Her Work In Solving $12 \ \textless \ -3(x+1$\] Contains 2 Common Errors Students Make When Solving Inequalities. Which Steps Contain The Errors? Explain What Jess Did

MP3 Essential Question: Common Errors in Solving Inequalities

Understanding the Problem

When solving inequalities, students often make common errors that can lead to incorrect solutions. In this article, we will explore two common errors made by students when solving inequalities, using Jess's work as an example. We will examine each step of her solution and identify the errors she made.

Jess's Work

Jess was given the inequality $12 \ \textless \ -3(x+1)$ and was asked to solve for $x$. Her work is as follows:

1. Distributing the negative 3

Jess started by distributing the negative 3 to the terms inside the parentheses:

$$-3(x+1) = -3x - 3$$

This step is correct, as Jess correctly applied the distributive property.

2. Adding 12 to both sides

Jess then added 12 to both sides of the inequality:

$$-3x - 3 + 12 \ \textless \ 12 + 12$$

This step is also correct, as Jess correctly added 12 to both sides of the inequality.

3. Simplifying the left side

Jess then simplified the left side of the inequality:

$$-3x + 9 \ \textless \ 24$$

This step is correct, as Jess correctly simplified the left side of the inequality.

4. Dividing both sides by -3

Jess then divided both sides of the inequality by -3:

$$\frac{-3x + 9}{-3} \ \textless \ \frac{24}{-3}$$

This is where Jess made her first error. When dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. In this case, the inequality sign should be reversed from $\textless$ to $\textgreater$.

5. Simplifying the left side

Jess then simplified the left side of the inequality:

$$x - 3 \ \textgreater \ -8$$

This step is correct, as Jess correctly simplified the left side of the inequality.

6. Adding 3 to both sides

Jess then added 3 to both sides of the inequality:

$$x - 3 + 3 \ \textgreater \ -8 + 3$$

This step is correct, as Jess correctly added 3 to both sides of the inequality.

7. Simplifying the left side

Jess then simplified the left side of the inequality:

$$x \ \textgreater \ -5$$

This step is correct, as Jess correctly simplified the left side of the inequality.

Common Errors in Solving Inequalities

From Jess's work, we can see that she made two common errors when solving inequalities:

1. Reversing the direction of the inequality sign when dividing by a negative number

When dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. In this case, the inequality sign should be reversed from $\textless$ to $\textgreater$.

2. Not checking the direction of the inequality sign

Jess's work shows that she did not check the direction of the inequality sign after dividing both sides by -3. This is a common error made by students when solving inequalities.

Conclusion

In conclusion, Jess's work contains two common errors made by students when solving inequalities. The first error is reversing the direction of the inequality sign when dividing by a negative number. The second error is not checking the direction of the inequality sign. By understanding these common errors, students can avoid making the same mistakes and arrive at the correct solution.

Real-World Applications

Solving inequalities is an essential skill in mathematics, with real-world applications in fields such as economics, finance, and engineering. In economics, inequalities are used to model the behavior of economic systems, while in finance, they are used to calculate interest rates and investment returns. In engineering, inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.

Tips for Solving Inequalities

To avoid making common errors when solving inequalities, follow these tips:

1. Check the direction of the inequality sign

When dividing both sides of an inequality by a negative number, reverse the direction of the inequality sign. 2. Simplify the inequality

Simplify the inequality by combining like terms and eliminating any unnecessary variables. 3. Check the solution

Check the solution by plugging in a value for the variable and verifying that the inequality is true.

Common Inequalities

Some common inequalities that students may encounter include:

• Linear inequalities: Inequalities of the form $ax + b \ \textless \ c$ or $ax + b \ \textgreater \ c$, where $a$, $b$, and $c$ are constants.
• Quadratic inequalities: Inequalities of the form $ax^2 + bx + c \ \textless \ 0$ or $ax^2 + bx + c \ \textgreater \ 0$, where $a$, $b$, and $c$ are constants.
• Absolute value inequalities: Inequalities of the form $|x| \ \textless \ a$ or $|x| \ \textgreater \ a$, where $a$ is a constant.

Conclusion

In conclusion, solving inequalities is an essential skill in mathematics, with real-world applications in fields such as economics, finance, and engineering. By understanding common errors and following tips for solving inequalities, students can avoid making mistakes and arrive at the correct solution.
MP3 Essential Question: Common Errors in Solving Inequalities - Q&A

Understanding the Problem

When solving inequalities, students often make common errors that can lead to incorrect solutions. In this article, we will explore two common errors made by students when solving inequalities, using Jess's work as an example. We will examine each step of her solution and identify the errors she made.

Q&A

Q: What are the two common errors made by students when solving inequalities?

A: The two common errors made by students when solving inequalities are:

1. Reversing the direction of the inequality sign when dividing by a negative number
2. Not checking the direction of the inequality sign

Q: Why is it important to check the direction of the inequality sign?

A: When dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. If you do not check the direction of the inequality sign, you may arrive at an incorrect solution.

Q: What is the correct way to solve an inequality?

A: To solve an inequality, follow these steps:

1. Check the direction of the inequality sign
2. Simplify the inequality
3. Check the solution

Q: What are some common inequalities that students may encounter?

A: Some common inequalities that students may encounter include:

• Linear inequalities: Inequalities of the form $ax + b \ \textless \ c$ or $ax + b \ \textgreater \ c$, where $a$, $b$, and $c$ are constants.
• Quadratic inequalities: Inequalities of the form $ax^2 + bx + c \ \textless \ 0$ or $ax^2 + bx + c \ \textgreater \ 0$, where $a$, $b$, and $c$ are constants.
• Absolute value inequalities: Inequalities of the form $|x| \ \textless \ a$ or $|x| \ \textgreater \ a$, where $a$ is a constant.

Q: How can I avoid making common errors when solving inequalities?

A: To avoid making common errors when solving inequalities, follow these tips:

1. Check the direction of the inequality sign
2. Simplify the inequality
3. Check the solution

Q: What are some real-world applications of solving inequalities?

A: Solving inequalities has many real-world applications, including:

• Economics: Inequalities are used to model the behavior of economic systems.
• Finance: Inequalities are used to calculate interest rates and investment returns.
• Engineering: Inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: How can I practice solving inequalities?

A: To practice solving inequalities, try the following:

• Work through examples: Practice solving inequalities by working through examples.
• Use online resources: Use online resources, such as video tutorials and practice problems, to help you practice solving inequalities.
• Ask for help: Ask your teacher or tutor for help if you are struggling to solve inequalities.

Conclusion

In conclusion, solving inequalities is an essential skill in mathematics, with real-world applications in fields such as economics, finance, and engineering. By understanding common errors and following tips for solving inequalities, students can avoid making mistakes and arrive at the correct solution.