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 , which is equivalent to 6 < 3 x .
'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 Inequality We are given the algebraic statement 6 < x ÷ 3 , which can be rewritten as 6 < 3 x . We need to determine which of the provided statements are equivalent to this inequality.
Analyzing Each Statement Let's analyze each statement:
"6 is at most some number, x , divided by 3." This translates to 6 ≤ 3 x , which is not equivalent.
"6 is less than the quotient of some number, x , and 3." This translates to 6 < 3 x , which is equivalent.
"6 is less than some number, x , divided by 3." This translates to 6 < 3 x , which is equivalent.
"6 is at most the quotient of some number, x , and 3." This translates to 6 ≤ 3 x , which is not equivalent.
"6 is less than the quotient of 3 and some number, x ." This translates to 6 < x 3 , which is not equivalent.
"6 is at most the product of some number, x , and 3." This translates to 6 ≤ 3 x , which is not equivalent.
"6 is less than the product of some number, x , and 3." This translates to 6 < 3 x , which is not equivalent.
"6 is less than 3 divided by some number, x ." This translates to 6 < x 3 , which is not equivalent.
"6 is at most the quotient of 3 and some number, x ." This translates to 6 ≤ x 3 , which is not equivalent.
"6 is at least some number, x , divided by 3." This translates to 6 ≥ 3 x , which 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.
Final Answer The equivalent statements 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 example, when determining the minimum score needed on a test to achieve a certain grade, or when calculating the minimum amount of ingredients required for a recipe. In this case, the inequality 6 < x /3 could represent a situation where you need to divide a certain amount of food ( x ) among 3 people, and each person needs to receive more than 6 units of food. This concept helps in resource allocation and planning.
The equivalent statements to 6 < x ÷ 3 are B and C, indicating that 6 is less than both the quotient of some number x and 3, and some number x divided by 3.
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