To determine how many solutions exist for the equation [tex]\(6 - 3x = 4 - x - 3 - 2x\)[/tex], we will follow these steps:
1. Simplify both sides of the equation:
Start by combining like terms on the right-hand side of the equation:
[tex]\[
4 - x - 3 - 2x = 4 - 3 - x - 2x = 1 - 3x
\][/tex]
Now, rewrite the equation with the simplified right-hand side:
[tex]\[
6 - 3x = 1 - 3x
\][/tex]
2. Isolate the variable [tex]\(x\)[/tex]:
To isolate the variable [tex]\(x\)[/tex], move all terms involving [tex]\(x\)[/tex] to one side of the equation and constant terms to the opposite side. However, in this case, both sides already contain the term [tex]\(-3x\)[/tex].
Subtract [tex]\(-3x\)[/tex] from both sides of the equation:
[tex]\[
6 - 3x + 3x = 1 - 3x + 3x
\][/tex]
Simplifying both sides:
[tex]\[
6 = 1
\][/tex]
3. Analyze the resulting statement:
The resulting equation [tex]\(6 = 1\)[/tex] is a false statement. This indicates an inconsistency, meaning there is no value of [tex]\(x\)[/tex] that can satisfy the original equation.
Therefore, the original equation [tex]\(6 - 3x = 4 - x - 3 - 2x\)[/tex] has no solutions. The correct answer is:
D. No solutions
