Solve for x: 1 + (-x - 6) / 12 = 3 + (4 + 8x) / 28

Steps to Solve

1. Rewrite the equation The given equation is $1 + \frac{-x - 6}{12} = 3 + \frac{4 + 8x}{28}$

2. Simplify the fractions Simplify the fraction $\frac{4 + 8x}{28}$ by dividing both numerator and denominator by 4. This gives $\frac{1+2x}{7}$. $1 + \frac{-x - 6}{12} = 3 + \frac{1 + 2x}{7}$

3. Eliminate fractions by multiplying by the least common multiple (LCM) The LCM of 12 and 7 is 84. Multiply both sides of the equation by 84. $84(1 + \frac{-x - 6}{12}) = 84(3 + \frac{1 + 2x}{7})$

4. Distribute and simplify Distribute the 84 on both sides: $84(1) + 84(\frac{-x - 6}{12}) = 84(3) + 84(\frac{1 + 2x}{7})$ $84 + 7(-x - 6) = 252 + 12(1 + 2x)$

5. Expand the parentheses $84 - 7x - 42 = 252 + 12 + 24x$

6. Combine like terms Combine the constants on the left side: $84 - 42 = 42$. Combine the constants on the right side: $252 + 12 = 264$. $42 - 7x = 264 + 24x$

7. Isolate x terms Add $7x$ to both sides: $42 - 7x + 7x = 264 + 24x + 7x$ $42 = 264 + 31x$

8. Isolate the constant terms Subtract 264 from both sides: $42 - 264 = 264 - 264 + 31x$ $-222 = 31x$

9. Solve for x Divide both sides by 31: $\frac{-222}{31} = \frac{31x}{31}$ $x = -\frac{222}{31}$ $x = - \frac{6 \times 37}{31}$