Solve $8x + 4 \geq 52$.A. $x \geq 7$ B. $x \geq 11$ C. $x \geq 2.5$ D. $x \geq 6$

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Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear inequality, which is a linear inequality in one variable. We will use the given problem, 8x+4≥528x + 4 \geq 52, as an example to demonstrate the step-by-step process of solving linear inequalities.

What are Linear Inequalities?

A linear inequality is an inequality that involves a linear expression, which is an expression that can be written in the form ax+bax + b, where aa and bb are constants, and xx is the variable. Linear inequalities can be written in the form ax+b≥cax + b \geq c, where cc is a constant. In this case, the inequality is 8x+4≥528x + 4 \geq 52.

Step 1: Subtract 4 from Both Sides

To solve the inequality, we need to isolate the variable xx. The first step is to subtract 4 from both sides of the inequality. This will give us:

8x+4−4≥52−48x + 4 - 4 \geq 52 - 4

Simplifying the left-hand side, we get:

8x≥488x \geq 48

Step 2: Divide Both Sides by 8

Next, we need to divide both sides of the inequality by 8. This will give us:

8x8≥488\frac{8x}{8} \geq \frac{48}{8}

Simplifying the left-hand side, we get:

x≥6x \geq 6

Conclusion

Therefore, the solution to the inequality 8x+4≥528x + 4 \geq 52 is x≥6x \geq 6. This means that any value of xx that is greater than or equal to 6 will satisfy the inequality.

Answer

The correct answer is:

  • D. x≥6x \geq 6

Why is this the Correct Answer?

This is the correct answer because we have isolated the variable xx and found that it must be greater than or equal to 6 in order to satisfy the inequality. This is the only value of xx that will make the inequality true.

Tips and Tricks

Here are some tips and tricks to help you solve linear inequalities:

  • Make sure to follow the order of operations (PEMDAS) when simplifying the inequality.
  • Use inverse operations to isolate the variable.
  • Check your work by plugging in a value of xx that satisfies the inequality to make sure it is true.

Real-World Applications

Linear inequalities have many real-world applications, such as:

  • Budgeting: A person may have a budget of $100 per week, and they want to know how much they can spend on entertainment.
  • Time management: A student may have a certain amount of time to complete a project, and they want to know how much time they can spend on each task.
  • Science: A scientist may have a certain amount of data, and they want to know how to analyze it.

Conclusion

In conclusion, solving linear inequalities is a crucial skill for students to master. By following the step-by-step process outlined in this article, you can solve linear inequalities with ease. Remember to make sure to follow the order of operations, use inverse operations to isolate the variable, and check your work by plugging in a value of xx that satisfies the inequality. With practice and patience, you will become proficient in solving linear inequalities and be able to apply them to real-world problems.

References

Additional Resources

Introduction

In our previous article, we discussed how to solve linear inequalities in one variable. In this article, we will provide a Q&A guide to help you better understand the concept of solving linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression, which is an expression that can be written in the form ax+bax + b, where aa and bb are constants, and xx is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable xx. You can do this by using inverse operations to get rid of the coefficient of xx and then solving for xx.

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 using inverse operations to isolate the variable
  • Not checking your work by plugging in a value of xx that satisfies the inequality

Q: How do I know if my solution is correct?

A: To check if your solution is correct, you need to plug in a value of xx that satisfies the inequality and make sure it is true. If it is true, then your solution is correct.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities. However, you need to make sure that you are using the correct function and that you are following the correct steps.

Q: Are linear inequalities used in real-world applications?

A: Yes, linear inequalities are used in many real-world applications, such as budgeting, time management, and science.

Q: Can I use linear inequalities to solve systems of equations?

A: Yes, you can use linear inequalities to solve systems of equations. However, you need to make sure that you are using the correct method and that you are following the correct steps.

Q: What are some common types of linear inequalities?

A: Some common types of linear inequalities include:

  • Linear inequalities in one variable
  • Linear inequalities in two variables
  • Quadratic inequalities

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to use a number line and plot the points that satisfy the inequality. You can use a solid line to represent the inequality and a dashed line to represent the boundary.

Q: Can I use linear inequalities to solve optimization problems?

A: Yes, you can use linear inequalities to solve optimization problems. However, you need to make sure that you are using the correct method and that you are following the correct steps.

Conclusion

In conclusion, solving linear inequalities is a crucial skill for students to master. By following the step-by-step process outlined in this article, you can solve linear inequalities with ease. Remember to make sure to follow the order of operations, use inverse operations to isolate the variable, and check your work by plugging in a value of xx that satisfies the inequality. With practice and patience, you will become proficient in solving linear inequalities and be able to apply them to real-world problems.

References

Additional Resources