Solve The Inequality:${ -\frac{3}{10}x - 7 \ \textless \ \frac{1}{2} }$

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Introduction

In this article, we will delve into the world of inequalities and focus on solving a specific one. The given inequality is −310x−7<12-\frac{3}{10}x - 7 < \frac{1}{2}. We will break down the solution process into manageable steps, making it easier to understand and follow along. By the end of this article, you will have a clear understanding of how to solve this type of inequality.

Understanding the Inequality

Before we dive into the solution, let's take a closer look at the inequality. The given inequality is −310x−7<12-\frac{3}{10}x - 7 < \frac{1}{2}. This is a linear inequality, which means it can be solved using basic algebraic operations.

Key Components of the Inequality

  • The coefficient of xx is −310-\frac{3}{10}.
  • The constant term is −7-7.
  • The inequality sign is less than (<<).
  • The right-hand side is 12\frac{1}{2}.

Step 1: Add 7 to Both Sides

To isolate the term containing xx, we need to get rid of the constant term on the left-hand side. We can do this by adding 7 to both sides of the inequality.

# Given inequality
inequality = "-3/10*x - 7 < 1/2"

new_inequality = "-3/10*x < 1/2 + 7"



simplified_inequality = "-3/10*x < 15/2"

Step 2: Simplify the Right-Hand Side

Now that we have added 7 to both sides, we can simplify the right-hand side by finding a common denominator.

# Simplify the right-hand side
simplified_inequality = "-3/10*x < 15/2"


common_denominator = 10



simplified_inequality = "-3/10*x < 75/10"

Step 3: Multiply Both Sides by -10

To get rid of the fraction on the left-hand side, we can multiply both sides of the inequality by -10. However, we need to be careful when multiplying by a negative number, as it will flip the direction of the inequality sign.

# Multiply both sides by -10
new_inequality = "3*x > -75"


flipped_inequality = "3*x > -75"

Step 4: Divide Both Sides by 3

Finally, we can divide both sides of the inequality by 3 to solve for xx.

# Divide both sides by 3
new_inequality = "x > -25"


simplified_inequality = "x > -25"

Conclusion

In this article, we have solved the inequality −310x−7<12-\frac{3}{10}x - 7 < \frac{1}{2} using basic algebraic operations. We added 7 to both sides, simplified the right-hand side, multiplied both sides by -10, and finally divided both sides by 3 to solve for xx. The solution to the inequality is x>−25x > -25.

Final Answer

The final answer is x>−25\boxed{x > -25}.

Related Topics

  • Solving linear inequalities
  • Basic algebraic operations
  • Inequality signs and their meanings

References

Note: The references provided are for educational purposes only and are not affiliated with the content of this article.

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Introduction

In our previous article, we solved the inequality −310x−7<12-\frac{3}{10}x - 7 < \frac{1}{2} using basic algebraic operations. In this article, we will provide a Q&A guide to help you better understand the solution process and address any questions you may have.

Q&A

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

A: The first step in solving the inequality is to add 7 to both sides to get rid of the constant term on the left-hand side.

Q: Why do we need to add 7 to both sides?

A: We need to add 7 to both sides to isolate the term containing xx. This will allow us to simplify the inequality and solve for xx.

Q: What happens when we multiply both sides by -10?

A: When we multiply both sides by -10, we need to flip the direction of the inequality sign. This is because multiplying by a negative number changes the direction of the inequality.

Q: Why do we need to divide both sides by 3?

A: We need to divide both sides by 3 to solve for xx. This will give us the final solution to the inequality.

Q: What is the final solution to the inequality?

A: The final solution to the inequality is x>−25x > -25.

Q: Can you provide an example of how to solve a similar inequality?

A: Let's consider the inequality 2x+5<102x + 5 < 10. To solve this inequality, we can subtract 5 from both sides to get 2x<52x < 5. Then, we can divide both sides by 2 to get x<52x < \frac{5}{2}.

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

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

  • Not flipping the direction of the inequality sign when multiplying by a negative number
  • Not simplifying the right-hand side of the inequality
  • Not isolating the term containing xx

Conclusion

In this article, we have provided a Q&A guide to help you better understand the solution process for the inequality −310x−7<12-\frac{3}{10}x - 7 < \frac{1}{2}. We have addressed common questions and provided examples to help you practice solving similar inequalities.

Final Tips

  • Make sure to follow the order of operations when solving inequalities
  • Simplify the right-hand side of the inequality as much as possible
  • Isolate the term containing xx to get the final solution

Related Topics

  • Solving linear inequalities
  • Basic algebraic operations
  • Inequality signs and their meanings

References

Note: The references provided are for educational purposes only and are not affiliated with the content of this article.