Let's analyze Logan's solution and reasoning step-by-step.
Step 1:
[tex]\[
\frac{3}{2}x - 10 = 50
\][/tex]
Reason 1: Given.
The initial equation is provided correctly.
Step 2:
[tex]\[
\frac{3}{2}x - 10 + 10 = 50 + 10
\][/tex]
Reason 2: Addition Property of Equality.
Here, Logan adds 10 to both sides of the equation to eliminate the -10 on the left side, which is correct.
Step 3:
[tex]\[
\frac{3}{2}x = 60
\][/tex]
Reason 3: Simplify.
By simplifying both sides of the equation after addition, this step is also correct.
Step 4:
[tex]\[
\frac{3}{2}x \left(\frac{2}{3}\right) = 60 \left(\frac{2}{3}\right)
\][/tex]
Reason 4: Division Property of Equality.
Here, Logan multiplies both sides of the equation by the reciprocal of [tex]\(\frac{3}{2}\)[/tex], which is [tex]\(\frac{2}{3}\)[/tex], to isolate [tex]\(x\)[/tex]. However, the proper naming for this step should be the Multiplication Property of Equality, not the Division Property of Equality. Multiplying by the reciprocal is a form of multiplication, not division.
Step 5:
[tex]\[
x = 40
\][/tex]
Reason 5: Simplify.
Simplifying the result, [tex]\(x = 40\)[/tex] is correct.
Thus, the incorrect reason is in Step 4. The error is that this should be stated as utilizing the Multiplication Property of Equality, not the Division Property of Equality.
The incorrect reason is:
c Reason 4
