To identify the mistake in solving the equation [tex]\(6x - 1 = -2x + 9\)[/tex], let's go through each step methodically and check for correctness and proper justifications:
1. Starting equation:
[tex]\(6x - 1 = -2x + 9\)[/tex]
2. Combining like terms:
Adding [tex]\(2x\)[/tex] to both sides of the equation to combine like terms on one side.
[tex]\[
6x - 1 + 2x = -2x + 9 + 2x
\][/tex]
Simplifies to:
[tex]\[
8x - 1 = 9
\][/tex]
The justification for this step is the Addition property of equality, which is correctly applied.
3. Removing the constant term from one side:
Adding 1 to both sides to isolate the term involving `x`.
[tex]\[
8x - 1 + 1 = 9 + 1
\][/tex]
Simplifies to:
[tex]\[
8x = 10
\][/tex]
The justification for this step is the Addition property of equality, which is correctly applied.
4. Solving for [tex]\(x\)[/tex]:
Dividing both sides by 8 to solve for `x`.
[tex]\[
\frac{8x}{8} = \frac{10}{8}
\][/tex]
Simplifies to:
[tex]\[
x = \frac{10}{8}
\][/tex]
The justification for this step is the Division property of equality, which is correctly applied.
5. Simplifying the fraction:
Simplifying the fraction:
[tex]\[
x = \frac{10}{8} = \frac{5}{4}
\][/tex]
Given the process, we notice an inconsistency in the provided solution steps. In step 4, the simplification results in:
[tex]\[
x = \frac{10}{8}
\][/tex]
However, there's a mistake in stating the intermediate result directly as:
[tex]\[
x = \frac{8}{10}
\][/tex]
This is incorrect since it should be [tex]\(x = \frac{10}{8}\)[/tex].
Therefore, the correct answer that identifies the mistake is:
C. Step 3 is incorrect and should be [tex]\( x = \frac{10}{8} \)[/tex].
