Solve The Inequality:$\[ \frac{11}{16}c - \frac{1}{6} \leq \frac{7}{12}c + \frac{5}{8} \\]

by ADMIN 91 views

=====================================================

Introduction

In this article, we will delve into the world of inequalities and learn how to solve a specific inequality involving fractions. Inequalities are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various mathematical disciplines, including algebra, calculus, and statistics. In this discussion, we will focus on solving the inequality 1116c−16≤712c+58\frac{11}{16}c - \frac{1}{6} \leq \frac{7}{12}c + \frac{5}{8}.

Understanding the Inequality

Before we dive into solving the inequality, let's first understand what it represents. The given inequality is a linear inequality, which means it can be represented in the form ax+b≤cx+dax + b \leq cx + d, where aa, bb, cc, and dd are constants, and xx is the variable. In this case, the variable is cc, and the constants are the fractions 1116\frac{11}{16}, 16\frac{1}{6}, 712\frac{7}{12}, and 58\frac{5}{8}.

Step 1: Simplify the Inequality

To solve the inequality, we need to simplify it by getting rid of the fractions. We can do this by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators, which is 4848. This will eliminate the fractions and make it easier to work with.

1116c−16≤712c+58\frac{11}{16}c - \frac{1}{6} \leq \frac{7}{12}c + \frac{5}{8}

Multiplying both sides by 4848:

48×1116c−48×16≤48×712c+48×5848 \times \frac{11}{16}c - 48 \times \frac{1}{6} \leq 48 \times \frac{7}{12}c + 48 \times \frac{5}{8}

Simplifying:

33c−8≤28c+3033c - 8 \leq 28c + 30

Step 2: Isolate the Variable

Now that we have simplified the inequality, we need to isolate the variable cc. We can do this by subtracting 28c28c from both sides of the inequality and then adding 88 to both sides.

33c−28c≤30+833c - 28c \leq 30 + 8

Simplifying:

5c≤385c \leq 38

Step 3: Solve for the Variable

Now that we have isolated the variable cc, we can solve for it by dividing both sides of the inequality by 55.

5c5≤385\frac{5c}{5} \leq \frac{38}{5}

Simplifying:

c≤385c \leq \frac{38}{5}

Conclusion

In this article, we have learned how to solve the inequality 1116c−16≤712c+58\frac{11}{16}c - \frac{1}{6} \leq \frac{7}{12}c + \frac{5}{8}. We simplified the inequality by multiplying both sides by the LCM of the denominators, isolated the variable cc by subtracting 28c28c from both sides and adding 88 to both sides, and finally solved for cc by dividing both sides of the inequality by 55. The solution to the inequality is c≤385c \leq \frac{38}{5}.

Final Answer

The final answer to the inequality is c≤385\boxed{c \leq \frac{38}{5}}.

Frequently Asked Questions

Q: What is the least common multiple (LCM) of the denominators?

A: The LCM of the denominators is 4848.

Q: How do I simplify the inequality?

A: To simplify the inequality, multiply both sides by the LCM of the denominators.

Q: How do I isolate the variable?

A: To isolate the variable, subtract 28c28c from both sides of the inequality and then add 88 to both sides.

Q: How do I solve for the variable?

A: To solve for the variable, divide both sides of the inequality by 55.

Additional Resources

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

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

Conclusion

In conclusion, solving inequalities is an essential skill in mathematics, and understanding how to solve them is crucial for success in various mathematical disciplines. In this article, we have learned how to solve the inequality 1116c−16≤712c+58\frac{11}{16}c - \frac{1}{6} \leq \frac{7}{12}c + \frac{5}{8} by simplifying it, isolating the variable, and solving for it. We hope this article has provided you with a better understanding of how to solve inequalities and has given you the confidence to tackle more complex mathematical problems.

=====================================

Introduction

In our previous article, we learned how to solve the inequality 1116c−16≤712c+58\frac{11}{16}c - \frac{1}{6} \leq \frac{7}{12}c + \frac{5}{8}. However, we understand that solving inequalities can be a challenging task, and many of you may have questions about the process. In this article, we will address some of the most frequently asked questions about solving inequalities.

Q&A

Q: What is the least common multiple (LCM) of the denominators?

A: The LCM of the denominators is the smallest number that all the denominators can divide into evenly. In the case of the inequality 1116c−16≤712c+58\frac{11}{16}c - \frac{1}{6} \leq \frac{7}{12}c + \frac{5}{8}, the LCM of the denominators is 4848.

Q: How do I simplify the inequality?

A: To simplify the inequality, multiply both sides by the LCM of the denominators. This will eliminate the fractions and make it easier to work with.

Q: What if the inequality has multiple variables?

A: If the inequality has multiple variables, you will need to isolate each variable separately. This may involve multiplying both sides of the inequality by the LCM of the denominators multiple times.

Q: Can I use a calculator to solve inequalities?

A: Yes, you can use a calculator to solve inequalities. However, it's always a good idea to check your work by hand to make sure you understand the process.

Q: What if the inequality has a negative sign?

A: If the inequality has a negative sign, you will need to flip the direction of the inequality. For example, if the inequality is −x≤5-x \leq 5, you would flip the direction of the inequality to get x≥−5x \geq -5.

Q: Can I use the same method to solve quadratic inequalities?

A: No, you cannot use the same method to solve quadratic inequalities. Quadratic inequalities require a different approach, and you will need to use a different method to solve them.

Q: What if I get stuck on a problem?

A: If you get stuck on a problem, don't be afraid to ask for help. You can ask a teacher, tutor, or classmate for assistance. You can also try breaking the problem down into smaller steps or looking for online resources to help you understand the concept.

Tips and Tricks

Tip 1: Read the problem carefully

Before you start solving the inequality, make sure you read the problem carefully and understand what is being asked.

Tip 2: Use a pencil and paper

It's always a good idea to work out the problem on a piece of paper with a pencil. This will help you keep track of your work and make it easier to check your answers.

Tip 3: Check your work

Make sure to check your work by plugging in a value for the variable and making sure the inequality holds true.

Tip 4: Practice, practice, practice

The more you practice solving inequalities, the more comfortable you will become with the process.

Conclusion

Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to read the problem carefully, use a pencil and paper, check your work, and practice, practice, practice. If you have any more questions or need further clarification, don't hesitate to ask.

Additional Resources

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

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

Final Answer

The final answer to the inequality is c≤385\boxed{c \leq \frac{38}{5}}.