Solve the equation 3[x - 6 - (2 - x)] + 1 = -[-(-2 + 6x)].

Step-by-step solution

1. 1

   Clear the innermost bracket on the left. Subtracting a bracket flips every sign inside it: $x-6-(2-x)=x-6-2+x=2x-8$. Working outward from the innermost bracket is what keeps the signs straight.

2. 2

   Multiply out and simplify the left side. $3(2x-8)+1=6x-24+1=6x-23$.

3. 3

   Unwrap the two minus signs on the right. Innermost first: $-(-2+6x)=2-6x$. Then the outer minus gives $-(2-6x)=-2+6x=6x-2$.

4. 4

   Compare the two sides. The equation is now $6x-23=6x-2$. Both sides have the same coefficient of $x$, so subtracting $6x$ eliminates the variable entirely and leaves $-23=-2$.

5. 5

   Interpret the false statement. A variable-free statement that is false means no value of $x$ can ever satisfy the equation — the two sides describe parallel lines with the same slope $6$ but different intercepts, so they never meet. The equation is inconsistent.

6. 6

   Sanity-check with a test value. At $x=0$ the left side is $3(0-6-2)+1=-23$ and the right side is $-2$; at $x=5$ the left side is $3(5-6+5-2)+1=7$ and the right side is $28$. The gap stays exactly $21$ for every $x$, confirming the sides never coincide.