Which Of These Inequalities Are Equivalent To $r \ \textgreater \ -11$? Check All That Apply.A. − R \textless 11 -r \ \textless \ 11 − R \textless 11 B. 3 Π \textless − 33 3\pi \ \textless \ -33 3 Π \textless − 33 C. 3 R \textgreater − 33 3r \ \textgreater \ -33 3 R \textgreater − 33 D. $-3r \ \textless \

Introduction

In mathematics, inequalities are a fundamental concept used to compare values and express relationships between variables. When solving inequalities, it's essential to understand the properties of equality and inequality operations. In this article, we will explore the concept of equivalent inequalities and how to identify them.

Understanding Equivalent Inequalities

Equivalent inequalities are statements that have the same solution set. In other words, if two inequalities are equivalent, they will have the same values that satisfy them. To determine if two inequalities are equivalent, we need to check if they have the same solution set.

Properties of Inequality Operations

Before we dive into solving inequalities, let's review the properties of inequality operations:

• Multiplication and Division: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign changes.
• Addition and Subtraction: When adding or subtracting a value from both sides of an inequality, the direction of the inequality sign remains the same.

Solving the Given Inequality

The given inequality is:

$$r \ \textgreater \ -11$$

We need to find equivalent inequalities that have the same solution set.

Option A: $-r \ \textless \ 11$

To check if this inequality is equivalent to the given inequality, let's multiply both sides of the given inequality by -1:

$$-r \ \textless \ -(-11)$$

$$-r \ \textless \ 11$$

This inequality is indeed equivalent to the given inequality.

Option B: $3\pi \ \textless \ -33$

This inequality is not equivalent to the given inequality. The variable $r$ is not present in this inequality, and the value of $\pi$ is not relevant to the given inequality.

Option C: $3r \ \textgreater \ -33$

To check if this inequality is equivalent to the given inequality, let's divide both sides of the given inequality by 3:

$$\frac{r}{3} \ \textgreater \ \frac{-11}{3}$$

$$r \ \textgreater \ -\frac{11}{3}$$

This inequality is not equivalent to the given inequality.

Option D: $-3r \ \textless \ 33$

To check if this inequality is equivalent to the given inequality, let's multiply both sides of the given inequality by -3:

$$-3r \ \textless \ -(-33)$$

$$-3r \ \textless \ 33$$

This inequality is indeed equivalent to the given inequality.

Conclusion

In conclusion, the equivalent inequalities to the given inequality $r \ \textgreater \ -11$ are:

• $-r \ \textless \ 11$
• $-3r \ \textless \ 33$

Q&A: Solving Inequalities

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

A: An equation is a statement that says two expressions are equal, while an inequality is a statement that says one expression is greater than, less than, or equal to another expression.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by a value, as long as you do the same operation to both sides.

Q: What happens when I multiply or divide both sides of an inequality by a negative number?

A: When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes. For example, if you have the inequality $x \ \textgreater \ 5$ and you multiply both sides by -1, the inequality becomes $-x \ \textless \ -5$.

Q: How do I determine if two inequalities are equivalent?

A: Two inequalities are equivalent if they have the same solution set. To determine if two inequalities are equivalent, you need to check if they have the same values that satisfy them.

Q: What is the difference between a linear inequality and a nonlinear inequality?

A: A linear inequality is an inequality that can be written in the form $ax \ \textless \ b$ or $ax \ \textgreater \ b$, where $a$ and $b$ are constants and $x$ is the variable. A nonlinear inequality is an inequality that cannot be written in this form.

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

A: To graph a linear inequality on a number line, you need to plot a point on the number line that satisfies the inequality and then draw an arrow in the direction of the inequality sign. For example, if you have the inequality $x \ \textgreater \ 2$, you would plot a point at $x = 2$ and draw an arrow to the right.

Q: What is the solution set of an inequality?

A: The solution set of an inequality is the set of all values that satisfy the inequality. For example, if you have the inequality $x \ \textgreater \ 2$, the solution set is all values greater than 2.

Q: How do I find the solution set of an inequality?

A: To find the solution set of an inequality, you need to solve the inequality and then write the solution in interval notation. For example, if you have the inequality $x \ \textgreater \ 2$, the solution set is $(2, \infty)$.

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

A: A strict inequality is an inequality that is written with a strict inequality sign, such as $x \ \textgreater \ 2$ or $x \ \textless \ 2$. A nonstrict inequality is an inequality that is written with a nonstrict inequality sign, such as $x \ \geq \ 2$ or $x \ \leq \ 2$.

Q: How do I solve a system of linear inequalities?

A: To solve a system of linear inequalities, you need to find the solution set of each inequality and then find the intersection of the solution sets. For example, if you have the system of inequalities $x \ \textgreater \ 2$ and $x \ \textless \ 5$, the solution set is $(2, 5)$.

Conclusion

In conclusion, solving inequalities is an essential skill in mathematics. By understanding the properties of inequality operations and how to solve inequalities, you can solve a wide range of problems in mathematics and other fields.