Which of the following statements is equivalent to the algebraic statement below? Select all that apply.
[tex]$6\ \textless \ x \div 3$[/tex]
A. 6 is at most some number, [tex]$x$[/tex], divided by 3.
B. 6 is less than the quotient of some number, [tex]$x$[/tex], and 3.
C. 6 is less than some number, [tex]$x$[/tex], divided by 3.
D. 6 is at most the quotient of some number, [tex]$x$[/tex], and 3.
E. 6 is less than the quotient of 3 and some number, [tex]$x$[/tex].
F. 6 is at most the product of some number, [tex]$x$[/tex], and 3.
G. 6 is less than the product of some number, [tex]$x$[/tex], and 3.
H. 6 is less than 3 divided by some number, [tex]$x$[/tex].
I. 6 is at most the quotient of 3 and some number, [tex]$x$[/tex].
J. 6 is at least some number, [tex]$x$[/tex], divided by 3.
Answer (2)
The algebraic statement is 6 < x ÷ 3 .
'6 is less than the quotient of some number, x , and 3' translates to 6 < 3 x .
'6 is less than some number, x , divided by 3' translates to 6 < 3 x .
The equivalent statements are: '6 is less than the quotient of some number, x , and 3' and '6 is less than some number, x , divided by 3'.
Explanation
Understanding the Problem We need to identify the statements that are equivalent to the algebraic statement 6 < x ÷ 3 . This is the same as saying that 6 is less than x divided by 3.
Analyzing Each Statement Let's analyze each statement:
"6 is at most some number, x , divided by 3" translates to 6 ≤ 3 x . This is not equivalent.
"6 is less than the quotient of some number, x , and 3" translates to 6 < 3 x . This is equivalent.
"6 is less than some number, x , divided by 3" translates to 6 < 3 x . This is equivalent.
"6 is at most the quotient of some number, x , and 3" translates to 6 ≤ 3 x . This is not equivalent.
"6 is less than the quotient of 3 and some number, x " translates to 6 < x 3 . This is not equivalent.
"6 is at most the product of some number, x , and 3" translates to 6 ≤ 3 x . This is not equivalent.
"6 is less than the product of some number, x , and 3" translates to 6 < 3 x . This is not equivalent.
"6 is less than 3 divided by some number, x " translates to 6 < x 3 . This is not equivalent.
"6 is at most the quotient of 3 and some number, x " translates to 6 ≤ x 3 . This is not equivalent.
"6 is at least some number, x , divided by 3" translates to 6 ≥ 3 x . This is not equivalent.
Identifying Equivalent Statements Therefore, the statements that are equivalent to 6 < x ÷ 3 are:
6 is less than the quotient of some number, x , and 3.
6 is less than some number, x , divided by 3.
Examples
Understanding inequalities is crucial in various real-life scenarios. For instance, when calculating the minimum score needed on a test to achieve a certain grade, or when determining the maximum amount of ingredients to stay within a budget. In this case, the inequality 6 < x /3 could represent a minimum requirement. For example, if you need to divide x amount of food among 3 people and each person needs to get more than 6 units of food, this inequality helps define the minimum amount of food required.
The equivalent statements to 6 < x ÷ 3 are: '6 is less than the quotient of some number, x , and 3' (B) and '6 is less than some number, x , divided by 3' (C). Other statements either state at most or reference different relationships, making them not equivalent. Thus, the correct options are B and C.
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