Yvonne needs to compare two expressions 1/3(9x + 6) and -(3x - 10) + 6(x - 1). She needs to know if one expression is greater than the other for all values of x. Which statement accurately describes the relationship between the two expressions.
A. The value of 1/3 (9x + 6) is always less than the value of - (3x - 10) + 6(x - 1)
B. The value of 1/3 (9x + 6) is always greater than the value of - (3x - 10) + 6(x - 1)
C. The value of 1/3 (9x + 6) is always equal to the value of - (3x - 10) + 6(x - 1)
D. The value of 1/3 (9x + 6) is sometimes greater than and sometimes less than the value of - (3x - 10) + 6(x - 1)
To solve this problem, we need to simplify both expressions and compare them.
The first expression, 1/3(9x + 6), can be simplified by distributing the 1/3 to both terms inside the parentheses: (1/3) * 9x + (1/3) * 6. This gives us 3x + 2.
The second expression, -(3x - 10) + 6(x - 1), can be simplified by distributing the -1 and 6 to the terms inside the parentheses: -3x + 10 + 6x - 6. Combining like terms, this gives us 3x + 4.
Comparing the simplified expressions, we have 3x + 2 versus 3x + 4.
These expressions have the same coefficient for x, which means that the value of the expression will only depend on the constant term.
Since 2 is always less than 4, no matter the value of x, we can conclude that:
A. The value of 1/3 (9x + 6) is always less than the value of - (3x - 10) + 6(x - 1).
To compare the two expressions, let's simplify them step-by-step.
Expression 1: 1/3(9x + 6)
To simplify, distribute 1/3 to both terms inside the parentheses:
1/3 * 9x + 1/3 * 6
This becomes:
3x + 2
Expression 2: -(3x - 10) + 6(x - 1)
To simplify, first distribute the negative sign:
-3x + 10 + 6(x - 1)
Next, distribute 6 to both terms inside the parentheses:
-3x + 10 + 6x - 6
Combine like terms:
3x + 4
Now, we can see that the simplified forms of the two expressions are 3x + 2 and 3x + 4.
To determine the relationship between the two expressions, we need to compare their coefficients of x. Both expressions have the same coefficient of x, which is 3. Therefore, the value of 3x + 4 will always be greater than the value of 3x + 2.
Therefore, the correct statement is:
A. The value of 1/3 (9x + 6) is always less than the value of - (3x - 10) + 6(x - 1)
To compare the expressions 1/3(9x + 6) and -(3x - 10) + 6(x - 1), we can simplify and analyze them.
Let's start with the first expression: 1/3(9x + 6)
To simplify it, we can distribute the 1/3 to both terms inside the parentheses:
1/3 * 9x + 1/3 * 6 = 3x + 2
Now let's simplify the second expression: -(3x - 10) + 6(x - 1)
To simplify it, we can distribute -1 and 6 to the terms inside the parentheses:
-1 * 3x + -1 * -10 + 6 * x + 6 * -1 = -3x + 10 + 6x - 6 = 3x + 4
Now, we have simplified both expressions. We can observe that the two expressions are 3x + 2 and 3x + 4.
Since the coefficient of x is positive in both expressions and the constant term is also positive in both expressions, we can conclude that the value of 3x + 4 is always greater than the value of 3x + 2.
Therefore, the statement that accurately describes the relationship between the two expressions is:
B. The value of 1/3 (9x + 6) is always greater than the value of - (3x - 10) + 6(x - 1)