1. \( \frac{x-2}{3}-\frac{2 x+1}{2}=4 \)

To solve the equation \( \frac{x-2}{3}-\frac{2x+1}{2}=4 \), first, let's eliminate the fractions by finding a common denominator, which is 6 in this case. Multiplying every term by 6 gives us: \[ 6 \left( \frac{x-2}{3} \right) - 6 \left( \frac{2x+1}{2} \right) = 6 \cdot 4 \] This simplifies to: \[ 2(x - 2) - 3(2x + 1) = 24 \] Expanding both terms: \[ 2x - 4 - 6x - 3 = 24 \] Combining like terms results in: \[ -4x - 7 = 24 \] Now, add 7 to both sides: \[ -4x = 31 \] Finally, divide by -4: \[ x = -\frac{31}{4} \] So, the solution to the equation is \( x = -\frac{31}{4} \).