Solve For $x$.$3x + 3 - X + (-7) \ \textgreater \ 6$A) $x \ \textless \ 5$ B) $x \ \textgreater \ 5$ C) $x \ \textgreater \ -5$ D) $x \ \textgreater \ 2.5$

by ADMIN 171 views

Introduction

In mathematics, solving linear inequalities is a crucial skill that helps us understand and analyze various real-world problems. A linear inequality is an inequality that can be written in the form of ax + b > c, where a, b, and c are constants, and x is the variable. In this article, we will focus on solving linear inequalities of the form ax + b > c, with a specific example: 3x + 3 - x + (-7) > 6.

Understanding the Problem

The given inequality is 3x + 3 - x + (-7) > 6. To solve this inequality, we need to simplify the left-hand side by combining like terms. Let's start by simplifying the expression:

3x - x + 3 - 7 > 6

Combine like terms:

2x - 4 > 6

Simplifying the Inequality

Now that we have simplified the inequality, we can isolate the variable x by adding 4 to both sides of the inequality:

2x > 10

Isolating the Variable

To isolate the variable x, we need to divide both sides of the inequality by 2:

x > 5

Analyzing the Solution

Now that we have solved the inequality, let's analyze the solution. The solution x > 5 means that x is greater than 5. This is a one-sided inequality, which means that x can take any value greater than 5, but not equal to 5.

Comparing the Solution to the Answer Choices

Let's compare the solution x > 5 to the answer choices:

A) x < 5 B) x > 5 C) x > -5 D) x > 2.5

The correct answer is B) x > 5.

Conclusion

Solving linear inequalities is an essential skill in mathematics that helps us analyze and understand various real-world problems. In this article, we solved the inequality 3x + 3 - x + (-7) > 6 by simplifying the left-hand side, isolating the variable x, and analyzing the solution. We also compared the solution to the answer choices and found that the correct answer is B) x > 5.

Tips and Tricks

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

  • Simplify the left-hand side of the inequality by combining like terms.
  • Isolate the variable x by adding or subtracting the same value from both sides of the inequality.
  • Divide both sides of the inequality by a non-zero value to isolate the variable x.
  • Analyze the solution to determine the correct answer choice.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving linear inequalities:

  • Failing to simplify the left-hand side of the inequality.
  • Not isolating the variable x by adding or subtracting the same value from both sides of the inequality.
  • Dividing both sides of the inequality by a zero value, which is undefined.
  • Not analyzing the solution to determine the correct answer choice.

Real-World Applications

Linear inequalities have many real-world applications, including:

  • Finance: Linear inequalities can be used to model financial problems, such as determining the minimum amount of money needed to invest in a stock.
  • Science: Linear inequalities can be used to model scientific problems, such as determining the minimum amount of time needed to complete a experiment.
  • Engineering: Linear inequalities can be used to model engineering problems, such as determining the minimum amount of material needed to build a structure.

Conclusion

Introduction

In our previous article, we discussed how to solve linear inequalities of the form ax + b > c. In this article, we will provide a Q&A guide to help you understand and apply the concepts of solving linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of ax + b > c, where a, b, and c are constants, and x is the variable.

Q: How do I simplify a linear inequality?

A: To simplify a linear inequality, combine like terms on the left-hand side by adding or subtracting the same value from both sides of the inequality.

Q: How do I isolate the variable x in a linear inequality?

A: To isolate the variable x, add or subtract the same value from both sides of the inequality, and then divide both sides by a non-zero value.

Q: What is the difference between a one-sided inequality and a two-sided inequality?

A: A one-sided inequality has a single inequality sign (>, <, ≄, or ≀), while a two-sided inequality has two inequality signs (e.g., 2x - 4 > 6 and 2x - 4 < 6).

Q: How do I analyze the solution to a linear inequality?

A: To analyze the solution, determine the correct answer choice by comparing the solution to the answer choices.

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

A: Some common mistakes to avoid include:

  • Failing to simplify the left-hand side of the inequality.
  • Not isolating the variable x by adding or subtracting the same value from both sides of the inequality.
  • Dividing both sides of the inequality by a zero value, which is undefined.
  • Not analyzing the solution to determine the correct answer choice.

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

A: Linear inequalities have many real-world applications, including:

  • Finance: Linear inequalities can be used to model financial problems, such as determining the minimum amount of money needed to invest in a stock.
  • Science: Linear inequalities can be used to model scientific problems, such as determining the minimum amount of time needed to complete an experiment.
  • Engineering: Linear inequalities can be used to model engineering problems, such as determining the minimum amount of material needed to build a structure.

Q: How can I practice solving linear inequalities?

A: You can practice solving linear inequalities by:

  • Working through example problems in your textbook or online resources.
  • Creating your own practice problems and solving them.
  • Joining a study group or working with a tutor to practice solving linear inequalities.

Conclusion

Solving linear inequalities is an essential skill in mathematics that helps us analyze and understand various real-world problems. By following the tips and tricks and avoiding common mistakes, you can become proficient in solving linear inequalities and apply them to real-world problems. Remember to practice regularly and seek help when needed to improve your skills.

Additional Resources

For more information on solving linear inequalities, check out the following resources:

  • Khan Academy: Linear Inequalities
  • Mathway: Linear Inequalities
  • Wolfram Alpha: Linear Inequalities

Final Tips

  • Always simplify the left-hand side of the inequality by combining like terms.
  • Isolate the variable x by adding or subtracting the same value from both sides of the inequality.
  • Analyze the solution to determine the correct answer choice.
  • Practice regularly to improve your skills in solving linear inequalities.