Which Inequality Is True For All Real Numbers? Check All That Apply.A. A − B \textless A + B A-b \ \textless \ A+b A − B \textless A + B B. A C ≥ B C Ac \geq Bc A C ≥ B C C. If A ≥ B A \geq B A ≥ B , Then A + C ≥ B + C A+c \geq B+c A + C ≥ B + C D. If C \textgreater D C \ \textgreater \ D C \textgreater D , Then $a-c \ \textless

Introduction

In mathematics, inequalities are used to compare the values of different expressions. They are an essential part of mathematical reasoning and problem-solving. In this article, we will explore four different inequalities and determine which ones are true for all real numbers.

A. $a-b \ \textless \ a+b$

Understanding the Inequality

The inequality $a-b \ \textless \ a+b$ can be rewritten as $a-b-a-b \ \textless \ 0$, which simplifies to $-2b \ \textless \ 0$. This inequality is true when $b$ is negative, but it is not true for all real numbers. For example, if $b=0$, the inequality becomes $0 \ \textless \ 0$, which is false.

Conclusion

The inequality $a-b \ \textless \ a+b$ is not true for all real numbers.

B. $ac \geq bc$

Understanding the Inequality

The inequality $ac \geq bc$ can be rewritten as $ac-bc \geq 0$, which simplifies to $c(a-b) \geq 0$. This inequality is true when $c$ is positive and $a-b$ is non-negative, or when $c$ is negative and $a-b$ is non-positive. However, it is not true for all real numbers. For example, if $c=0$, the inequality becomes $0 \geq 0$, which is true, but it is not true for all real numbers.

Conclusion

The inequality $ac \geq bc$ is not true for all real numbers.

C. If $a \geq b$, then $a+c \geq b+c$

Understanding the Inequality

The inequality $a+c \geq b+c$ is a statement that is true for all real numbers. If $a \geq b$, then adding the same value $c$ to both sides of the inequality will not change the direction of the inequality. This is because the addition of $c$ is a common term that can be subtracted from both sides of the inequality, leaving the original inequality intact.

Conclusion

The inequality "If $a \geq b$, then $a+c \geq b+c$" is true for all real numbers.

D. If $c \ \textgreater \ d$, then $a-c \ \textless \ a-d$

Understanding the Inequality

The inequality $a-c \ \textless \ a-d$ can be rewritten as $a-c-a+d \ \textless \ 0$, which simplifies to $d-c \ \textless \ 0$. This inequality is true when $c$ is greater than $d$, but it is not true for all real numbers. For example, if $c=d$, the inequality becomes $0 \ \textless \ 0$, which is false.

Conclusion

The inequality "If $c \ \textgreater \ d$, then $a-c \ \textless \ a-d$" is not true for all real numbers.

Conclusion

In conclusion, the only inequality that is true for all real numbers is "If $a \geq b$, then $a+c \geq b+c$". This inequality is a fundamental property of inequalities and is used extensively in mathematics.

Frequently Asked Questions

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

A: An inequality is a statement that compares the values of two expressions, while an equation is a statement that sets two expressions equal to each other.

Q: How do I determine if an inequality is true for all real numbers?

A: To determine if an inequality is true for all real numbers, you can try substituting different values for the variables and see if the inequality holds true.

Q: What is the importance of inequalities in mathematics?

A: Inequalities are used extensively in mathematics to compare the values of different expressions. They are an essential part of mathematical reasoning and problem-solving.

Final Thoughts

In conclusion, inequalities are an essential part of mathematics and are used extensively in mathematical reasoning and problem-solving. The inequality "If $a \geq b$, then $a+c \geq b+c$" is the only one that is true for all real numbers. We hope that this article has provided a clear understanding of inequalities and their importance in mathematics.

Introduction

In our previous article, we explored four different inequalities and determined which ones are true for all real numbers. In this article, we will answer some frequently asked questions about inequalities and provide additional information to help you better understand these mathematical concepts.

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

A: An inequality is a statement that compares the values of two expressions, while an equation is a statement that sets two expressions equal to each other. For example, the statement "x > 2" is an inequality, while the statement "x + 2 = 4" is an equation.

Q: How do I determine if an inequality is true for all real numbers?

A: To determine if an inequality is true for all real numbers, you can try substituting different values for the variables and see if the inequality holds true. For example, if you are trying to determine if the inequality "a - b < a + b" is true for all real numbers, you can try substituting different values for a and b, such as a = 1 and b = 2, or a = 0 and b = 0.

Q: What is the importance of inequalities in mathematics?

A: Inequalities are used extensively in mathematics to compare the values of different expressions. They are an essential part of mathematical reasoning and problem-solving. Inequalities are used to solve problems in a wide range of fields, including algebra, geometry, calculus, and statistics.

Q: How do I solve an inequality?

A: To solve an inequality, you can use a variety of techniques, including:

• Adding or subtracting the same value to both sides of the inequality
• Multiplying or dividing both sides of the inequality by the same non-zero value
• Using inverse operations to isolate the variable
• Graphing the inequality on a number line

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict symbol, such as "<" or ">", while a non-strict inequality is an inequality that is written with a non-strict symbol, such as "≤" or "≥". For example, the statement "x < 2" is a strict inequality, while the statement "x ≤ 2" is a non-strict inequality.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you can use the following steps:

• Draw a number line and mark the value that is being compared to the variable
• Shade the region to the left or right of the marked value, depending on the direction of the inequality
• Mark the endpoint of the inequality, if it is a non-strict inequality

Q: What is the difference between an open interval and a closed interval?

A: An open interval is an interval that is written with parentheses, such as (a, b), while a closed interval is an interval that is written with square brackets, such as [a, b]. For example, the statement "x ∈ (1, 2)" is an open interval, while the statement "x ∈ [1, 2]" is a closed interval.

Q: How do I determine if an inequality is true for a specific value?

A: To determine if an inequality is true for a specific value, you can substitute the value into the inequality and see if it holds true. For example, if you are trying to determine if the inequality "x > 2" is true for x = 3, you can substitute x = 3 into the inequality and see that it is true.

Q: What is the importance of inequalities in real-world applications?

A: Inequalities are used extensively in real-world applications, including:

• Finance: Inequalities are used to compare the values of different investments and to determine the best investment strategy.
• Science: Inequalities are used to compare the values of different physical quantities and to determine the best experimental design.
• Engineering: Inequalities are used to compare the values of different materials and to determine the best material for a specific application.

Conclusion

In conclusion, inequalities are an essential part of mathematics and are used extensively in mathematical reasoning and problem-solving. We hope that this article has provided a clear understanding of inequalities and their importance in mathematics.