For Numbers 18a-18f, Select True Or False For The Inequality.18a. 0.21 \textless 0.27 0.21 \ \textless \ 0.27 0.21 \textless 0.27 - True - False 18b. 0.4 \textgreater 0.45 0.4 \ \textgreater \ 0.45 0.4 \textgreater 0.45 - True - False 18c. $ \$ $ 3.21 \ \textgreater \ $ 0.2$ - True

Introduction

Inequalities are mathematical statements that compare two or more values, indicating whether they are greater than, less than, or equal to each other. In this article, we will explore the concept of inequalities and provide a step-by-step guide on how to solve them. We will also discuss the importance of inequalities in mathematics and their real-world applications.

What are Inequalities?

Inequalities are mathematical statements that compare two or more values using the following symbols:

• Greater than (>): $a > b$
• Less than (<): $a < b$
• Greater than or equal to (≥): $a ≥ b$
• Less than or equal to (≤): $a ≤ b$

For example, the statement $2x + 3 > 5$ is an inequality, where $2x + 3$ is greater than $5$.

Types of Inequalities

There are two main types of inequalities: linear and nonlinear.

• Linear Inequalities: These are inequalities that can be written in the form $ax + b > c$, where $a$, $b$, and $c$ are constants.
• Nonlinear Inequalities: These are inequalities that cannot be written in the form $ax + b > c$, where $a$, $b$, and $c$ are constants.

Solving Linear Inequalities

To solve a linear inequality, we need to isolate the variable on one side of the inequality sign. Here are the steps to follow:

1. Add or subtract the same value to both sides: If the inequality is of the form $ax + b > c$, we can add or subtract the same value to both sides to get $ax > c - b$.
2. Multiply or divide both sides by the same value: If the inequality is of the form $ax > c - b$, we can multiply or divide both sides by the same value to get $x > \frac{c - b}{a}$.

Solving Nonlinear Inequalities

To solve a nonlinear inequality, we need to use algebraic techniques such as factoring, quadratic formula, or graphing. Here are the steps to follow:

1. Factor the inequality: If the inequality is of the form $ax^2 + bx + c > 0$, we can factor the left-hand side to get $(x + d)(x + e) > 0$.
2. Use the quadratic formula: If the inequality is of the form $ax^2 + bx + c > 0$, we can use the quadratic formula to find the roots of the equation.
3. Graph the inequality: If the inequality is of the form $ax^2 + bx + c > 0$, we can graph the left-hand side to find the intervals where the inequality is true.

Real-World Applications of Inequalities

Inequalities have numerous real-world applications in fields such as economics, finance, and engineering. Here are a few examples:

• Budgeting: Inequalities can be used to compare the cost of different products or services.
• Investing: Inequalities can be used to compare the returns on different investments.
• Engineering: Inequalities can be used to design and optimize systems such as bridges, buildings, and electronic circuits.

Conclusion

Inequalities are an essential part of mathematics, and they have numerous real-world applications. By understanding and solving inequalities, we can make informed decisions in fields such as economics, finance, and engineering. In this article, we have provided a comprehensive guide on how to solve linear and nonlinear inequalities, and we have discussed the importance of inequalities in mathematics.

Practice Problems

Here are some practice problems to help you understand and solve inequalities:

18a. $0.21 \ \textless \ 0.27$

• True
• False

18b. $0.4 \ \textgreater \ 0.45$

• True
• False

18c. $\$$ 3.21 \ \textgreater \ $ 0.2$

• True
• False

Answer Key

18a. $0.21 \ \textless \ 0.27$

• False

18b. $0.4 \ \textgreater \ 0.45$

• False

18c. $\$$ 3.21 \ \textgreater \ $ 0.2$

• True

Discussion

In this article, we have discussed the concept of inequalities and provided a step-by-step guide on how to solve them. We have also discussed the importance of inequalities in mathematics and their real-world applications. If you have any questions or need further clarification, please feel free to ask in the discussion section below.

References

• [1] "Inequalities" by Khan Academy
• [2] "Linear and Nonlinear Inequalities" by Math Open Reference
• [3] "Real-World Applications of Inequalities" by Wolfram Alpha

Note

Introduction

In our previous article, we discussed the concept of inequalities and provided a step-by-step guide on how to solve them. In this article, we will answer some of the most frequently asked questions about inequalities.

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two or more values, indicating whether they are greater than, less than, or equal to each other.

Q: What are the different types of inequalities?

A: There are two main types of inequalities: linear and nonlinear.

• Linear Inequalities: These are inequalities that can be written in the form $ax + b > c$, where $a$, $b$, and $c$ are constants.
• Nonlinear Inequalities: These are inequalities that cannot be written in the form $ax + b > c$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. Here are the steps to follow:

1. Add or subtract the same value to both sides: If the inequality is of the form $ax + b > c$, you can add or subtract the same value to both sides to get $ax > c - b$.
2. Multiply or divide both sides by the same value: If the inequality is of the form $ax > c - b$, you can multiply or divide both sides by the same value to get $x > \frac{c - b}{a}$.

Q: How do I solve a nonlinear inequality?

A: To solve a nonlinear inequality, you need to use algebraic techniques such as factoring, quadratic formula, or graphing. Here are the steps to follow:

1. Factor the inequality: If the inequality is of the form $ax^2 + bx + c > 0$, you can factor the left-hand side to get $(x + d)(x + e) > 0$.
2. Use the quadratic formula: If the inequality is of the form $ax^2 + bx + c > 0$, you can use the quadratic formula to find the roots of the equation.
3. Graph the inequality: If the inequality is of the form $ax^2 + bx + c > 0$, you can graph the left-hand side to find the intervals where the inequality is true.

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

A: Inequalities have numerous real-world applications in fields such as economics, finance, and engineering. Here are a few examples:

• Budgeting: Inequalities can be used to compare the cost of different products or services.
• Investing: Inequalities can be used to compare the returns on different investments.
• Engineering: Inequalities can be used to design and optimize systems such as bridges, buildings, and electronic circuits.

Q: How do I determine if an inequality is true or false?

A: To determine if an inequality is true or false, you need to evaluate the expression on both sides of the inequality sign. If the expression on the left-hand side is greater than, less than, or equal to the expression on the right-hand side, then the inequality is true. Otherwise, the inequality is false.

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

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

• Not isolating the variable: Make sure to isolate the variable on one side of the inequality sign.
• Not checking the direction of the inequality: Make sure to check the direction of the inequality sign.
• Not considering the domain: Make sure to consider the domain of the inequality.

Conclusion

In this article, we have answered some of the most frequently asked questions about inequalities. We hope that this article has been helpful in clarifying any doubts you may have had about inequalities. If you have any further questions, please feel free to ask in the discussion section below.

Practice Problems

Here are some practice problems to help you understand and solve inequalities:

18d. $2x + 3 > 5$

• True
• False

18e. $x^2 + 4x + 4 > 0$

• True
• False

18f. $\$$ 2.50 \ \textless \ $ 3.00$

• True
• False

Answer Key

18d. $2x + 3 > 5$

• True

18e. $x^2 + 4x + 4 > 0$

• True

18f. $\$$ 2.50 \ \textless \ $ 3.00$

• True

Discussion

In this article, we have discussed some of the most frequently asked questions about inequalities. We hope that this article has been helpful in clarifying any doubts you may have had about inequalities. If you have any further questions, please feel free to ask in the discussion section below.

References

• [1] "Inequalities" by Khan Academy
• [2] "Linear and Nonlinear Inequalities" by Math Open Reference
• [3] "Real-World Applications of Inequalities" by Wolfram Alpha

Note

This article is for educational purposes only and is not intended to be a comprehensive guide on inequalities. If you need further clarification or have any questions, please feel free to ask in the discussion section below.