Sara Solved The Inequality $-4x + 1 \ \textgreater \ 9$ And Graphed The Solution Set. She Made A Mistake But Cannot Determine Where. Her Work Is Shown Below.Where Did Sara Make A Mistake? Choose The First Incorrect Step In Sara's Work.Given

Understanding the Problem

Solving and graphing inequalities can be a challenging task, especially when it comes to identifying mistakes. In this article, we will guide you through the process of solving and graphing the inequality $-4x + 1 > 9$ and identify the first incorrect step in Sara's work.

Step 1: Write Down the Inequality

The given inequality is $-4x + 1 > 9$. The first step is to write down the inequality and understand what it means.

Step 2: Subtract 1 from Both Sides

To isolate the term with the variable, we need to subtract 1 from both sides of the inequality.

$$-4x + 1 - 1 > 9 - 1$$

This simplifies to:

$$-4x > 8$$

Step 3: Divide Both Sides by -4

To solve for x, we need to divide both sides of the inequality by -4. However, when we divide or multiply an inequality by a negative number, we need to reverse the direction of the inequality sign.

$$\frac{-4x}{-4} < \frac{8}{-4}$$

This simplifies to:

$$x < -2$$

Step 4: Graph the Solution Set

To graph the solution set, we need to draw a number line and mark the point -2 as a closed circle, since it is included in the solution set.

Step 5: Check the Solution

To check the solution, we need to plug in a value of x that is less than -2 into the original inequality and verify that it is true.

Let's say we plug in x = -3.

$$-4(-3) + 1 > 9$$

This simplifies to:

$$12 + 1 > 9$$

Which is true.

Identifying the Mistake

Sara's work is shown below:

1. $-4x + 1 > 9$
2. $-4x > 8$
3. $x > -2$
4. Graph the solution set

The first incorrect step in Sara's work is step 3, where she incorrectly wrote $x > -2$ instead of $x < -2$.

Conclusion

Solving and graphing inequalities requires careful attention to detail and a thorough understanding of the steps involved. By following the steps outlined in this article, you can ensure that you are solving and graphing inequalities correctly. Remember to check your work and identify any mistakes to ensure that you are getting the correct solution.

Common Mistakes to Avoid

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

• Incorrectly writing the inequality sign: Make sure to write the inequality sign correctly, especially when dividing or multiplying by a negative number.
• Forgetting to reverse the inequality sign: When dividing or multiplying an inequality by a negative number, make sure to reverse the direction of the inequality sign.
• Not checking the solution: Always check the solution by plugging in a value of x that is in the solution set and verifying that it is true.

By avoiding these common mistakes, you can ensure that you are solving and graphing inequalities correctly and accurately.

Final Thoughts

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about solving and graphing inequalities.

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to write down the inequality and understand what it means. This involves identifying the variable, the inequality sign, and any constants.

Q: How do I isolate the variable in an inequality?

A: To isolate the variable in an inequality, you need to perform operations on both sides of the inequality to get the variable by itself. This may involve adding, subtracting, multiplying, or dividing both sides of the inequality.

Q: What happens when I divide or multiply an inequality by a negative number?

A: When you divide or multiply an inequality by a negative number, you need to reverse the direction of the inequality sign. For example, if you have the inequality $x > 5$ and you multiply both sides by -1, the resulting inequality would be $x < -5$.

Q: How do I graph the solution set of an inequality?

A: To graph the solution set of an inequality, you need to draw a number line and mark the point or points that are included in the solution set. If the inequality is of the form $x > a$, you would mark the point $a$ as an open circle. If the inequality is of the form $x < a$, you would mark the point $a$ as a closed circle.

Q: What is the difference between a closed circle and an open circle on a number line?

A: A closed circle on a number line represents a point that is included in the solution set, while an open circle represents a point that is not included in the solution set.

Q: How do I check the solution of an inequality?

A: To check the solution of an inequality, you need to plug in a value of x that is in the solution set and verify that it is true. For example, if you have the inequality $x > 5$ and you plug in x = 6, you would get $6 > 5$, which is true.

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

A: Some common mistakes to avoid when solving and graphing inequalities include:

• Incorrectly writing the inequality sign: Make sure to write the inequality sign correctly, especially when dividing or multiplying by a negative number.
• Forgetting to reverse the inequality sign: When dividing or multiplying an inequality by a negative number, make sure to reverse the direction of the inequality sign.
• Not checking the solution: Always check the solution by plugging in a value of x that is in the solution set and verifying that it is true.

Q: How can I practice solving and graphing inequalities?

A: You can practice solving and graphing inequalities by working through examples and exercises in a textbook or online resource. You can also try solving and graphing inequalities on your own and checking your work with a friend or teacher.

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

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

• Optimization problems: Solving and graphing inequalities can be used to optimize problems such as finding the maximum or minimum value of a function.
• Data analysis: Solving and graphing inequalities can be used to analyze data and make predictions about future trends.
• Engineering: Solving and graphing inequalities can be used to design and optimize systems such as bridges, buildings, and electronic circuits.

Conclusion

Solving and graphing inequalities is an important skill that has many real-world applications. By following the steps outlined in this article and practicing regularly, you can become proficient in solving and graphing inequalities. Remember to check your work and identify any mistakes to ensure that you are getting the correct solution.