Let's carefully analyze Noah's steps to solve the equation and point out why the 'divide each side by [tex]\( x \)[/tex]' step is unacceptable.
1. Original Equation:
[tex]\[
2(x + 6) - 4 = 8 + 6x
\][/tex]
2. Apply the distributive property:
[tex]\[
2(x + 6) - 4 \rightarrow 2x + 12 - 4
\][/tex]
So the equation becomes:
[tex]\[
2x + 12 - 4 = 8 + 6x
\][/tex]
3. Combine like terms:
Combine [tex]\(12 - 4\)[/tex]:
[tex]\[
2x + 8 = 8 + 6x
\][/tex]
4. Subtract 8 from both sides:
Subtract 8 from both sides to simplify:
[tex]\[
2x = 6x
\][/tex]
5. Problematic step - 'divide each side by [tex]\( x \)[/tex]':
Dividing both sides by [tex]\( x \)[/tex]:
[tex]\[
\frac{2x}{x} = \frac{6x}{x} \rightarrow 2 = 6
\][/tex]
This leads to an incorrect statement because 2 does not equal 6. This step is invalid because dividing by essentially zero (when [tex]\( x = 0 \)[/tex]) is undefined and can lead to incorrect conclusions.
Instead, let's look at a correct step to avoid the invalid operation:
Subtracting [tex]\( 2x \)[/tex] from both sides to isolate the variable terms:
[tex]\[
2x - 2x = 6x - 2x
\][/tex]
This simplifies to:
[tex]\[
0 = 4x
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Divide both sides by 4 to find [tex]\( x \)[/tex]:
[tex]\[
x = 0
\][/tex]
Thus, Noah's incorrect step of dividing by [tex]\( x \)[/tex] without considering [tex]\( x \)[/tex] could be zero leads to an incorrect equation [tex]\( 2 = 6 \)[/tex]. This is why the 'divide each side by [tex]\( x \)[/tex]' step is unacceptable: it could lead us to think that there is no solution while, in fact, the solution is [tex]\( x = 0 \)[/tex].
