Solve The Inequality:$-8s + 4s + 6 \ \textless \ 3s + 6$

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve a specific inequality. Inequalities are mathematical expressions that compare two values, and they are used to describe relationships between variables. Solving an inequality involves finding the values of the variable that satisfy the inequality. In this case, we will be solving the inequality −8s+4s+6 \textless 3s+6-8s + 4s + 6 \ \textless \ 3s + 6.

Understanding the Inequality

Before we start solving the inequality, let's first understand what it means. The inequality −8s+4s+6 \textless 3s+6-8s + 4s + 6 \ \textless \ 3s + 6 can be read as "negative 8s plus 4s plus 6 is less than 3s plus 6". This means that we need to find the values of s that make the left-hand side of the inequality less than the right-hand side.

Simplifying the Inequality

To simplify the inequality, we can start by combining like terms on the left-hand side. We have −8s+4s-8s + 4s, which can be combined to get −4s-4s. So, the inequality becomes −4s+6 \textless 3s+6-4s + 6 \ \textless \ 3s + 6.

Isolating the Variable

Next, we need to isolate the variable s on one side of the inequality. To do this, we can add 4s to both sides of the inequality. This will give us −4s+4s+6 \textless 3s+4s+6-4s + 4s + 6 \ \textless \ 3s + 4s + 6, which simplifies to 6 \textless 7s+66 \ \textless \ 7s + 6.

Solving for s

Now, we need to solve for s. To do this, we can subtract 6 from both sides of the inequality. This will give us 6−6 \textless 7s+6−66 - 6 \ \textless \ 7s + 6 - 6, which simplifies to 0 \textless 7s0 \ \textless \ 7s.

Final Solution

The final solution to the inequality is s \textgreater 0s \ \textgreater \ 0. This means that s must be greater than 0 in order to satisfy the inequality.

Conclusion

Solving an inequality involves finding the values of the variable that satisfy the inequality. In this case, we started with the inequality −8s+4s+6 \textless 3s+6-8s + 4s + 6 \ \textless \ 3s + 6 and simplified it to s \textgreater 0s \ \textgreater \ 0. This means that s must be greater than 0 in order to satisfy the inequality.

Tips and Tricks

  • When solving an inequality, it's essential to follow the order of operations (PEMDAS) to ensure that you're combining like terms correctly.
  • When adding or subtracting the same value to both sides of an inequality, you're not changing the direction of the inequality.
  • When multiplying or dividing both sides of an inequality by a negative value, you need to reverse the direction of the inequality.

Real-World Applications

Inequalities have many real-world applications, including:

  • Finance: Inequalities are used to calculate interest rates and investment returns.
  • Science: Inequalities are used to model population growth and chemical reactions.
  • Engineering: Inequalities are used to design and optimize systems.

Common Mistakes

  • Not following the order of operations (PEMDAS) when simplifying an inequality.
  • Not reversing the direction of the inequality when multiplying or dividing both sides by a negative value.
  • Not checking the solution to ensure that it satisfies the original inequality.

Final Thoughts

Solving an inequality involves finding the values of the variable that satisfy the inequality. In this case, we started with the inequality −8s+4s+6 \textless 3s+6-8s + 4s + 6 \ \textless \ 3s + 6 and simplified it to s \textgreater 0s \ \textgreater \ 0. This means that s must be greater than 0 in order to satisfy the inequality. By following the steps outlined in this article, you can solve any inequality that comes your way.

Additional Resources

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

  • Khan Academy: Inequalities
  • Mathway: Inequality Solver
  • Wolfram Alpha: Inequality Solver

Frequently Asked Questions

  • Q: What is an inequality?
  • A: An inequality is a mathematical expression that compares two values.
  • Q: How do I solve an inequality?
  • A: To solve an inequality, you need to find the values of the variable that satisfy the inequality.
  • Q: What are some common mistakes to avoid when solving an inequality?
  • A: Some common mistakes to avoid when solving an inequality include not following the order of operations (PEMDAS), not reversing the direction of the inequality when multiplying or dividing both sides by a negative value, and not checking the solution to ensure that it satisfies the original inequality.

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Introduction

In our previous article, we explored the world of inequalities and learned how to solve a specific inequality. In this article, we will continue to delve into the world of inequalities and answer some of the most frequently asked questions about solving inequalities.

Q&A

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values. It is used to describe relationships between variables and can be used to model real-world situations.

Q: How do I solve an inequality?

A: To solve an inequality, you need to find the values of the variable that satisfy the inequality. This involves simplifying the inequality, isolating the variable, and checking the solution to ensure that it satisfies the original inequality.

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

A: Some common mistakes to avoid when solving an inequality include:

  • Not following the order of operations (PEMDAS) when simplifying an inequality.
  • Not reversing the direction of the inequality when multiplying or dividing both sides by a negative value.
  • Not checking the solution to ensure that it satisfies the original inequality.

Q: How do I know if I have solved the inequality correctly?

A: To ensure that you have solved the inequality correctly, you need to check the solution to ensure that it satisfies the original inequality. This involves plugging the solution back into the original inequality and verifying that it is true.

Q: Can I use the same steps to solve all types of inequalities?

A: No, the steps for solving inequalities can vary depending on the type of inequality. For example, when solving a linear inequality, you can use the same steps as we outlined in our previous article. However, when solving a quadratic inequality, you may need to use a different approach.

Q: How do I know if an inequality is linear or quadratic?

A: To determine if an inequality is linear or quadratic, you need to look at the highest power of the variable. If the highest power is 1, then the inequality is linear. If the highest power is 2, then the inequality is quadratic.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality. However, you need to be careful when using a calculator to ensure that you are entering the correct values and that the calculator is set to the correct mode.

Q: How do I graph an inequality?

A: To graph an inequality, you need to use a number line or a coordinate plane. You can then shade the region that satisfies the inequality.

Q: Can I use a graphing calculator to graph an inequality?

A: Yes, you can use a graphing calculator to graph an inequality. This can be a useful tool for visualizing the solution to an inequality.

Conclusion

Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. By following the steps outlined in this article and avoiding common mistakes, you can ensure that you are solving inequalities correctly.

Tips and Tricks

  • When solving an inequality, it's essential to follow the order of operations (PEMDAS) to ensure that you're combining like terms correctly.
  • When adding or subtracting the same value to both sides of an inequality, you're not changing the direction of the inequality.
  • When multiplying or dividing both sides of an inequality by a negative value, you need to reverse the direction of the inequality.

Real-World Applications

Inequalities have many real-world applications, including:

  • Finance: Inequalities are used to calculate interest rates and investment returns.
  • Science: Inequalities are used to model population growth and chemical reactions.
  • Engineering: Inequalities are used to design and optimize systems.

Common Mistakes

  • Not following the order of operations (PEMDAS) when simplifying an inequality.
  • Not reversing the direction of the inequality when multiplying or dividing both sides by a negative value.
  • Not checking the solution to ensure that it satisfies the original inequality.

Final Thoughts

Solving inequalities involves finding the values of the variable that satisfy the inequality. By following the steps outlined in this article and avoiding common mistakes, you can ensure that you are solving inequalities correctly. Whether you are a student or a professional, inequalities are an essential part of mathematics and have many real-world applications.

Additional Resources

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

  • Khan Academy: Inequalities
  • Mathway: Inequality Solver
  • Wolfram Alpha: Inequality Solver

Frequently Asked Questions

  • Q: What is an inequality?
  • A: An inequality is a mathematical expression that compares two values.
  • Q: How do I solve an inequality?
  • A: To solve an inequality, you need to find the values of the variable that satisfy the inequality.
  • Q: What are some common mistakes to avoid when solving an inequality?
  • A: Some common mistakes to avoid when solving an inequality include not following the order of operations (PEMDAS), not reversing the direction of the inequality when multiplying or dividing both sides by a negative value, and not checking the solution to ensure that it satisfies the original inequality.