Which of the following statements is equivalent to the algebraic expression [tex]$13-8 \geq 4$[/tex]?
A. The subtraction of 8 from 13 is at least 4.
B. The subtraction of 8 from 13 is greater than or equal to 4.
C. The subtraction of 8 from 13 is at most 4.
D. 8 less than 13 is greater than or equal to 4.
E. 13 minus 8 is at most 4.
F. 13 minus 8 is greater than or equal to 4.
G. The subtraction of 13 from 8 is at least 4.
H. 13 less than 8 is at least 4.
I. 13 less than 8 is at most 4.
J. The subtraction of 13 from 8 is at most 4.
Answer (1)
Simplify the inequality: 13 − 8 g e 4 becomes 5 g e 4 , which is true.
Check each statement against the simplified inequality.
Identify the statements that are also true.
The equivalent statements are: 'The subtraction of 8 from 13 is at least 4', 'The subtraction of 8 from 13 is greater than or equal to 4', '8 less than 13 is greater than or equal to 4', '13 minus 8 is greater than or equal to 4', and '13 less than 8 is at most 4'.
Explanation
Simplifying the Inequality We are given the inequality 13 − 8 g e 4 . First, let's simplify the left side of the inequality: 13 − 8 = 5 . So the inequality becomes 5 g e 4 , which is true. Now, we need to identify which of the given statements are equivalent to this inequality.
Analyzing Each Statement Let's analyze each statement:
"The subtraction of 8 from 13 is at least 4": This means 13 − 8 g e 4 , which is 5 g e 4 . This is true.
"The subtraction of 8 from 13 is greater than or equal to 4": This also means 13 − 8 g e 4 , which is 5 g e 4 . This is true.
"The subtraction of 8 from 13 is at most 4": This means 13 − 8 l e 4 , which is 5 l e 4 . This is false.
"8 less than 13 is greater than or equal to 4": This means 13 − 8 g e 4 , which is 5 g e 4 . This is true.
"13 minus 8 is at most 4": This means 13 − 8 l e 4 , which is 5 l e 4 . This is false.
"13 minus 8 is greater than or equal to 4": This means 13 − 8 g e 4 , which is 5 g e 4 . This is true.
"The subtraction of 13 from 8 is at least 4": This means 8 − 13 g e 4 , which is − 5 g e 4 . This is false.
"13 less than 8 is at least 4": This means 8 − 13 g e 4 , which is − 5 g e 4 . This is false.
"13 less than 8 is at most 4": This means 8 − 13 l e 4 , which is − 5 l e 4 . This is true.
"The subtraction of 1 from 8 is at most 4": This statement is unrelated to the original inequality. It means 8 − 1 l e 4 , which is 7 l e 4 . This is false.
Identifying Equivalent Statements The statements that are equivalent to 13 − 8 g e 4 are the ones that are true. These are:
The subtraction of 8 from 13 is at least 4.
The subtraction of 8 from 13 is greater than or equal to 4.
8 less than 13 is greater than or equal to 4.
13 minus 8 is greater than or equal to 4.
13 less than 8 is at most 4.
Final Answer Therefore, the equivalent statements are:
The subtraction of 8 from 13 is at least 4
The subtraction of 8 from 13 is greater than or equal to 4
8 less than 13 is greater than or equal to 4
13 minus 8 is greater than or equal to 4
13 less than 8 is at most 4
Examples
Understanding inequalities is crucial in many real-life situations, such as budgeting. For example, if you have $13 and need to spend at least $8 on groceries, you want to ensure that you have at least $4 remaining. The inequality 13 − 8 g e 4 helps you confirm that you will have enough money left after buying the groceries. This concept extends to more complex financial planning, resource allocation, and decision-making processes where ensuring a minimum or maximum threshold is met is essential.
