Question

1. nora and so - hee each solve the equation ( 2 ( 6 - 4 x ) = - 2 x - 2 ) in different ways.

noras work
( 2 ( 6 - 4 x ) = - 2 x - 2 )
( 2 ( 6 ) - 4 x = - 2 x - 2 )
( 12 - 4 x = - 2 x - 2 )
( 12 - 4 x + 4 x = - 2 x + 4 x - 2 )
( 12 = 2 x - 2 )
( 12 + 2 = 2 x - 2 + 2 )
( 14 = 2 x )
( \frac { 1 } { 2 } ( 14 ) = \frac { 1 } { 2 } ( 2 x ) )
( 7 = x )
the solution is 7.
so - hees work
( 2 ( 6 - 4 x ) = - 2 x - 2 )
( \frac { 1 } { 2 } ( 2 ( 6 - 4 x ) ) = \frac { 1 } { 2 } ( - 2 x - 2 ) )
( 6 - 4 x = - x - 1 )
( 6 - 4 x + x = - x + x - 1 )
( 6 - 3 x = - 1 )
( 6 + ( - 6 ) - 3 x = - 1 + ( - 6 ) )
( - 3 x = - 7 )
( - \frac { 1 } { 3 } ( - 3 x ) = - \frac { 1 } { 3 } ( - 7 ) )
( x = \frac { 7 } { 3 } )
the solution is ( \frac { 7 } { 3 } ).
a. which students solution shows a correct method for solving the equation?
b. in the incorrect solution, circle the step that first shows an error.
c. explain why the step you circled is incorrect.

Explanation:

Step1: Analyze Nora's work

Nora made a mistake in the distributive property. The correct distributive property is \(a(b + c)=ab+ac\). In her case, \(2(6 - 4x)=2\times6-2\times4x = 12-8x\), but she wrote \(2(6)-4x\).

Step2: Analyze So - hee's work

So - hee's work:

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She correctly used the properties of equality (multiplication property of equality, addition property of equality) to solve the equation.

Step3: Answer part a

So - hee's solution shows a correct method for solving the equation.

Step4: Answer part b

In Nora's work, the step \(2(6)-4x=-2x - 2\) is the first incorrect step.

Step5: Answer part c

When using the distributive property \(a(b + c)=ab+ac\), for \(2(6 - 4x)\), it should be \(2\times6-2\times4x\) (i.e., \(12 - 8x\)), not \(2\times6-4x\). Nora did not multiply \(4x\) by \(2\) correctly.

Answer:

a. So - hee's solution is correct.
b. In Nora's work, the step \(2(6)-4x=-2x - 2\) is the first incorrect step.
c. When using the distributive property \(a(b + c)=ab + ac\) for \(2(6 - 4x)\), it should be \(2\times6-2\times4x\) (i.e., \(12 - 8x\)), not \(2\times6-4x\). Nora did not multiply \(4x\) by \(2\) correctly.