Here Is A True Statement: − 8.7 \textless − 8.4 -8.7 \ \textless \ -8.4 − 8.7 \textless − 8.4 . Select All Of The Statements That Are Equivalent To − 8.7 \textless − 8.4 -8.7 \ \textless \ -8.4 − 8.7 \textless − 8.4 .a. -8.7 Is Further To The Right On The Number Line Than -8.4.b. -8.7 Is Further To The Left On The

Mar 12, 2025 by ADMIN 320 views

Understanding Inequalities: A Closer Look at the Number Line

In mathematics, inequalities are used to compare the values of two or more numbers. When we say that one number is less than another, we are essentially comparing their positions on the number line. In this article, we will explore the concept of inequalities and how they relate to the number line. We will also examine the given statement $-8.7 \ \textless \ -8.4$ and determine which of the provided options are equivalent to it.

What is a Number Line?

A number line is a visual representation of the set of real numbers, with each point on the line corresponding to a unique number. The number line is typically drawn with the positive numbers on the right and the negative numbers on the left. The number line is a useful tool for understanding the relationships between numbers and for solving problems involving inequalities.

Understanding Inequalities

An inequality is a statement that compares the values of two or more numbers. Inequalities can be written in a variety of ways, including:

$a < b$ (read as "a is less than b")
$a > b$ (read as "a is greater than b")
$a \leq b$ (read as "a is less than or equal to b")
$a \geq b$ (read as "a is greater than or equal to b")

In the given statement $-8.7 \ \textless \ -8.4$ , we are comparing the values of two numbers, $-8.7$ and $-8.4$ . The statement is saying that $-8.7$ is less than $-8.4$ .

Option a: -8.7 is further to the right on the number line than -8.4

This option is incorrect because if $-8.7$ is further to the right on the number line than $-8.4$ , then it would be greater than $-8.4$ , not less than. On the number line, the number $-8.7$ is actually to the left of $-8.4$ .

Option b: -8.7 is further to the left on the number line than -8.4

This option is correct because if $-8.7$ is further to the left on the number line than $-8.4$ , then it would be less than $-8.4$ , which is consistent with the given statement $-8.7 \ \textless \ -8.4$ .

Option c: -8.7 is equal to -8.4

This option is incorrect because if $-8.7$ were equal to $-8.4$ , then the statement $-8.7 \ \textless \ -8.4$ would not be true.

Option d: -8.7 is greater than -8.4

This option is incorrect because if $-8.7$ were greater than $-8.4$ , then the statement $-8.7 \ \textless \ -8.4$ would not be true.

Conclusion

In conclusion, the only option that is equivalent to the given statement $-8.7 \ \textless \ -8.4$ is option b: $-8.7$ is further to the left on the number line than $-8.4$ . This option correctly reflects the relationship between the two numbers on the number line.

Additional Examples

Here are a few more examples of inequalities and how they relate to the number line:

$-5 \ \textless \ -3$ : This statement is true because $-5$ is further to the left on the number line than $-3$ .
$-2 \ \textless \ 0$ : This statement is true because $-2$ is further to the left on the number line than $0$ .
$0 \ \textless \ 2$ : This statement is true because $0$ is further to the left on the number line than $2$ .

Practice Problems

Here are a few practice problems to help you understand inequalities and the number line:

Which of the following statements is equivalent to $-6 \ \textless \ -4$ ? a. $-6$ is further to the right on the number line than $-4$ . b. $-6$ is further to the left on the number line than $-4$ . c. $-6$ is equal to $-4$ . d. $-6$ is greater than $-4$ .

Answer: b. $-6$ is further to the left on the number line than $-4$ .

Which of the following statements is equivalent to $-1 \ \textless \ 1$ ? a. $-1$ is further to the right on the number line than $1$ . b. $-1$ is further to the left on the number line than $1$ . c. $-1$ is equal to $1$ . d. $-1$ is greater than $1$ .

Answer: b. $-1$ is further to the left on the number line than $1$ .

Conclusion

In conclusion, inequalities are an important concept in mathematics that can be used to compare the values of two or more numbers. The number line is a useful tool for understanding the relationships between numbers and for solving problems involving inequalities. By understanding inequalities and the number line, you can solve a wide range of problems and become a more confident and proficient mathematician.
Frequently Asked Questions: Inequalities and the Number Line

In this article, we will answer some of the most frequently asked questions about inequalities and the number line.

Q: What is the difference between a number line and a coordinate plane?

A: A number line is a visual representation of the set of real numbers, with each point on the line corresponding to a unique number. A coordinate plane, on the other hand, is a two-dimensional grid that consists of a horizontal axis (x-axis) and a vertical axis (y-axis). While both number lines and coordinate planes are used to represent numbers, they serve different purposes and are used in different contexts.

Q: How do I determine if a number is positive or negative on the number line?

A: To determine if a number is positive or negative on the number line, look at the sign of the number. If the number has a positive sign (+), it is located to the right of zero on the number line. If the number has a negative sign (-), it is located to the left of zero on the number line.

Q: How do I compare two numbers on the number line?

A: To compare two numbers on the number line, look at their positions relative to each other. If one number is to the right of the other number, it is greater. If one number is to the left of the other number, it is less.

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

A: An inequality is a statement that compares the values of two or more numbers, but does not indicate whether they are equal or not. An equation, on the other hand, is a statement that indicates that two or more numbers are equal.

Q: How do I solve an inequality?

A: To solve an inequality, follow these steps:

Write the inequality in the form of an equation.
Solve the equation for the variable.
Check the solution to make sure it satisfies the original inequality.

Q: What is the concept of "greater than or equal to" (≥) and "less than or equal to" (≤)?

A: The concept of "greater than or equal to" (≥) and "less than or equal to" (≤) is used to indicate that a number is either greater than or equal to a certain value, or less than or equal to a certain value. For example, the statement "x ≥ 5" means that x is either greater than or equal to 5.

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

A: To graph an inequality on the number line, follow these steps:

Write the inequality in the form of an equation.
Solve the equation for the variable.
Plot the solution on the number line.
Use a closed circle to indicate that the solution is included, or an open circle to indicate that the solution is not included.

Q: What is the concept of "absolute value"?

A: The concept of "absolute value" is used to indicate the distance of a number from zero on the number line. For example, the absolute value of -3 is 3, because -3 is 3 units away from zero on the number line.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, follow these steps:

Write the inequality in the form of an equation.
Solve the equation for the variable.
Check the solution to make sure it satisfies the original inequality.

Conclusion

In conclusion, inequalities and the number line are fundamental concepts in mathematics that are used to compare the values of two or more numbers. By understanding these concepts, you can solve a wide range of problems and become a more confident and proficient mathematician.