Solve The Inequality: R 32 \textless 5 8 = 30 T \frac{r}{32} \ \textless \ \frac{5}{8} = \frac{30}{t} 32 R ​ \textless 8 5 ​ = T 30 ​

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve a specific one. The given inequality is r32 \textless 58=30t\frac{r}{32} \ \textless \ \frac{5}{8} = \frac{30}{t}. We will break down the solution into manageable steps, making it easy to understand and follow along. By the end of this article, you will be able to solve similar inequalities with confidence.

Understanding the Inequality

Before we dive into the solution, let's take a closer look at the inequality. We have two fractions: r32\frac{r}{32} and 58\frac{5}{8}. The inequality states that r32\frac{r}{32} is less than 58\frac{5}{8}. We also have a second fraction, 30t\frac{30}{t}, which is equal to 58\frac{5}{8}. Our goal is to solve for rr and tt.

Step 1: Simplify the Inequality

To simplify the inequality, we can start by cross-multiplying. This means multiplying both sides of the inequality by the denominators of the fractions. In this case, we will multiply both sides by 3232 and 88.

r32 \textless 58\frac{r}{32} \ \textless \ \frac{5}{8}

Cross-multiplying:

8r \textless 5×328r \ \textless \ 5 \times 32

Simplifying:

8r \textless 1608r \ \textless \ 160

Step 2: Solve for r

Now that we have simplified the inequality, we can solve for rr. To do this, we need to isolate rr on one side of the inequality.

8r \textless 1608r \ \textless \ 160

Dividing both sides by 88:

r \textless 1608r \ \textless \ \frac{160}{8}

Simplifying:

r \textless 20r \ \textless \ 20

Step 3: Solve for t

Now that we have solved for rr, we can turn our attention to solving for tt. We know that 30t=58\frac{30}{t} = \frac{5}{8}. We can use this equation to solve for tt.

30t=58\frac{30}{t} = \frac{5}{8}

Cross-multiplying:

30×8=5×t30 \times 8 = 5 \times t

Simplifying:

240=5t240 = 5t

Dividing both sides by 55:

t=2405t = \frac{240}{5}

Simplifying:

t=48t = 48

Conclusion

In this article, we solved the inequality r32 \textless 58=30t\frac{r}{32} \ \textless \ \frac{5}{8} = \frac{30}{t}. We broke down the solution into manageable steps, making it easy to understand and follow along. By the end of this article, you should be able to solve similar inequalities with confidence.

Key Takeaways

  • To solve an inequality, start by simplifying it using cross-multiplication.
  • Once you have simplified the inequality, solve for the variable by isolating it on one side of the inequality.
  • Use the equation 30t=58\frac{30}{t} = \frac{5}{8} to solve for tt.

Frequently Asked Questions

  • What is the value of rr?
    • r \textless 20r \ \textless \ 20
  • What is the value of tt?
    • t=48t = 48

Further Reading

  • Inequalities: A Comprehensive Guide
  • Algebra: A Step-by-Step Guide
  • Math: A Beginner's Guide

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 r32 \textless 58=30t\frac{r}{32} \ \textless \ \frac{5}{8} = \frac{30}{t}. We broke down the solution into manageable steps, making it easy to understand and follow along. In this article, we will answer some of the most frequently asked questions about solving inequalities.

Q&A

Q: What is the value of r?

A: The value of rr is r \textless 20r \ \textless \ 20. This means that rr is less than 20.

Q: What is the value of t?

A: The value of tt is t=48t = 48. This means that tt is equal to 48.

Q: How do I simplify an inequality?

A: To simplify an inequality, start by cross-multiplying. This means multiplying both sides of the inequality by the denominators of the fractions.

Q: How do I solve for a variable in an inequality?

A: To solve for a variable in an inequality, isolate the variable on one side of the inequality. This means getting the variable by itself on one side of the inequality.

Q: What is the difference between an inequality and an equation?

A: An inequality is a statement that two expressions are not equal, while an equation is a statement that two expressions are equal.

Q: Can I use the same steps to solve a linear inequality as I would to solve a quadratic inequality?

A: No, the steps to solve a linear inequality are different from the steps to solve a quadratic inequality. Linear inequalities involve one variable, while quadratic inequalities involve two variables.

Q: How do I know which direction to go when solving an inequality?

A: When solving an inequality, you need to determine the direction of the inequality. This means determining whether the inequality is less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).

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 make sure that the calculator is set to the correct mode (e.g., fraction mode) and that you are using the correct operations (e.g., division).

Conclusion

In this article, we answered some of the most frequently asked questions about solving inequalities. We covered topics such as simplifying inequalities, solving for variables, and determining the direction of the inequality. By following these steps and practicing with different types of inequalities, you will become more confident in your ability to solve inequalities.

Key Takeaways

  • To simplify an inequality, start by cross-multiplying.
  • To solve for a variable in an inequality, isolate the variable on one side of the inequality.
  • Determine the direction of the inequality by using the correct operations (e.g., division).
  • Use a calculator to solve an inequality, but make sure it is set to the correct mode (e.g., fraction mode).

Frequently Asked Questions

  • What is the value of r?
    • r \textless 20r \ \textless \ 20
  • What is the value of t?
    • t=48t = 48
  • How do I simplify an inequality?
    • Cross-multiply
  • How do I solve for a variable in an inequality?
    • Isolate the variable on one side of the inequality

Further Reading

  • Inequalities: A Comprehensive Guide
  • Algebra: A Step-by-Step Guide
  • Math: A Beginner's Guide

References