To determine if the expression \(4x - 3\) is equivalent to \(\frac{1}{2}(4x - 6) - 2x(1 - 3)\), Latesha evaluates both expressions by substituting \(x = 2\).
1. Evaluate the first expression \(4x - 3\) when \(x = 2\):
[tex]\[
4(2) - 3
\][/tex]
First, multiply 4 by 2:
[tex]\[
8
\][/tex]
Next, subtract 3:
[tex]\[
8 - 3 = 5
\][/tex]
2. Evaluate the second expression \(\frac{1}{2}(4x - 6) - 2x(1 - 3)\) when \(x = 2\):
[tex]\[
\frac{1}{2}(4(2) - 6) - 2(2)(1 - 3)
\][/tex]
First, compute the term inside the parentheses \(4(2) - 6\):
[tex]\[
8 - 6 = 2
\][/tex]
Then, compute \(\frac{1}{2} \times 2\):
[tex]\[
1
\][/tex]
Next, evaluate \(2(2)\) and the term inside the parentheses \(1 - 3\):
[tex]\[
2 \times 2 = 4
\][/tex]
[tex]\[
1 - 3 = -2
\][/tex]
Now, multiply \(-2\) to the result:
[tex]\[
4 \times -2 = -8
\][/tex]
Finally, combine the results from the two terms:
[tex]\[
1 - (-8) = 1 + 8 = 9
\][/tex]
Since Latesha gets 5 for the first expression and 9 for the second expression, she should compare the values:
[tex]\[5 \text{ and } 9\][/tex]
Therefore, Latesha should compare 5 and 9 to justify whether the expressions are equivalent.
Correct pairs: [tex]\(\boxed{5 \text{ and } 9}\)[/tex]
