Answered

2.9 Practice Problems
Possible Points: 11.11

Noah is solving an equation and one of his moves is unacceptable. Here are the moves he made.

[tex]\[
\begin{array}{cc}
2(x+6)-4 & = 8+6x \\
2x+12-4 & = 8+6x \\
2x+8 & = 8+6x \\
2x & = 6x \\
2 & = 6
\end{array}
\][/tex]

Original equation
Apply the distributive property
Combine like terms
Subtract 8 from both sides
Divide each side by [tex]\(x\)[/tex]

Which answer best explains why the "divide each side by [tex]\(x\)[/tex] step" is unacceptable?

A. When you divide both sides of [tex]\(2x = 6x\)[/tex] by [tex]\(x\)[/tex], you get [tex]\(2x^2 = 6x^2\)[/tex].
B. When you divide both sides of [tex]\(2x = 6x\)[/tex] by [tex]\(x\)[/tex], it could lead us to think that there is no solution while in fact the solution is [tex]\(x = 0\)[/tex].
C. When you divide both sides of [tex]\(2x = 6x\)[/tex] by [tex]\(x\)[/tex], you get [tex]\(2 = 6x\)[/tex].
D. No solution, while in fact the solution is [tex]\(x = 0\)[/tex].



Answer :

GinnyAnswer
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].