Michelle Says That The Solution To The Inequality $2(4y-3)\ \textgreater \ -22$ Is $y\ \textgreater \ -3.5$. Her Work Is Shown:$\[ \begin{aligned} 2(4y-3) & \ \textgreater \ -22 \\ 8y - 6 & \ \textgreater \ -22 \\ 8y & \

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. A linear inequality is an inequality that involves a linear expression, and it is often used to model real-world situations. In this article, we will focus on solving linear inequalities, with a specific example of the inequality $2(4y-3) > -22$. We will analyze the solution provided by Michelle and determine if it is correct.

Understanding the Inequality

The given inequality is $2(4y-3) > -22$. To solve this inequality, we need to follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses: $4y-3$
2. Multiply the result by 2: $2(4y-3)$
3. Compare the result to -22: $2(4y-3) > -22$

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is $4y-3$. To evaluate this expression, we need to distribute the 4 to the y and the -3:

$4y-3 = 4y - 3$

Step 2: Multiply the Result by 2

Now, we need to multiply the result by 2:

$2(4y-3) = 2(4y) - 2(3)$

Using the distributive property, we can simplify this expression:

$2(4y-3) = 8y - 6$

Step 3: Compare the Result to -22

Now, we can compare the result to -22:

$8y - 6 > -22$

Adding 6 to Both Sides

To isolate the term with the variable (8y), we need to add 6 to both sides of the inequality:

$8y - 6 + 6 > -22 + 6$

Simplifying the inequality, we get:

$8y > -16$

Dividing Both Sides by 8

To solve for y, we need to divide both sides of the inequality by 8:

$\frac{8y}{8} > \frac{-16}{8}$

Simplifying the inequality, we get:

$y > -2$

Conclusion

In conclusion, the solution to the inequality $2(4y-3) > -22$ is $y > -2$. Michelle's work is incorrect, as she stated that the solution is $y > -3.5$. The correct solution is $y > -2$.

Why is Michelle's Solution Incorrect?

Michelle's solution is incorrect because she did not follow the order of operations (PEMDAS) when solving the inequality. She also did not simplify the expression correctly, which led to an incorrect solution.

Tips for Solving Linear Inequalities

When solving linear inequalities, it is essential to follow the order of operations (PEMDAS) and simplify the expression correctly. Additionally, make sure to isolate the term with the variable and check the direction of the inequality.

Common Mistakes to Avoid

When solving linear inequalities, there are several common mistakes to avoid:

• Not following the order of operations (PEMDAS)
• Not simplifying the expression correctly
• Not isolating the term with the variable
• Not checking the direction of the inequality

Conclusion

In conclusion, solving linear inequalities requires careful attention to detail and a thorough understanding of the order of operations (PEMDAS). By following these steps and avoiding common mistakes, you can solve linear inequalities with confidence.

Final Answer

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Introduction

In our previous article, we discussed how to solve linear inequalities, with a specific example of the inequality $2(4y-3) > -22$. We analyzed the solution provided by Michelle and determined that it was incorrect. In this article, we will provide a Q&A guide to help you better understand how to solve linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression. It is often used to model real-world situations, such as the cost of a product or the time it takes to complete a task.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when solving an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, follow these steps:

1. Evaluate the expression inside any parentheses.
2. Multiply or divide both sides of the inequality by the same value.
3. Add or subtract the same value from both sides of the inequality.
4. Check the direction of the inequality.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is an equation that involves a linear expression, and it is often used to model real-world situations. A linear inequality, on the other hand, is an inequality that involves a linear expression. The key difference between the two is that a linear equation has an equal sign (=), while a linear inequality has an inequality sign (<, >, ≤, or ≥).

Q: How do I know which direction to use when solving a linear inequality?

A: When solving a linear inequality, you need to determine which direction to use. If the inequality is of the form $ax > b$, where $a$ and $b$ are positive numbers, then the solution is $x > \frac{b}{a}$. If the inequality is of the form $ax < b$, where $a$ and $b$ are positive numbers, then the solution is $x < \frac{b}{a}$. If the inequality is of the form $ax \geq b$, where $a$ and $b$ are positive numbers, then the solution is $x \geq \frac{b}{a}$. If the inequality is of the form $ax \leq b$, where $a$ and $b$ are positive numbers, then the solution is $x \leq \frac{b}{a}$.

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

A: Some common mistakes to avoid when solving linear inequalities include:

• Not following the order of operations (PEMDAS)
• Not simplifying the expression correctly
• Not isolating the term with the variable
• Not checking the direction of the inequality

Q: How do I check my work when solving a linear inequality?

A: To check your work when solving a linear inequality, follow these steps:

1. Write down the original inequality.
2. Write down the solution you found.
3. Check that the solution satisfies the original inequality.
4. Check that the solution is consistent with the direction of the inequality.

Conclusion

In conclusion, solving linear inequalities requires careful attention to detail and a thorough understanding of the order of operations (PEMDAS). By following these steps and avoiding common mistakes, you can solve linear inequalities with confidence.

Final Answer

The final answer is: $\boxed{y > -2}$