Solve for \( x \) : \( \begin{array}{ll}\text { (1) } 7+9 x=7 x-3 & \text { (2) }-9+2 x=6-x \\ \text { (3) }-6(1-x)=10-2 x & \text { (4) } 9(x-3)=5(x-1)-14 \\ \text { (5) } \frac{2}{3}(x+2)-\frac{x+1}{5}=3 & \text { (6) } \frac{x-1}{2}-\frac{x+6}{3}=-2 \\ \text { (7) } x(x-2)+1=(x-2)^{2} & \text { (8) } \\ (x+2)(x-2)=(x+3)^{2}\end{array} \)

**(1)** Solve \[ 7+9x=7x-3. \] Subtract \(7x\) from both sides: \[ 7+2x=-3. \] Subtract 7 from both sides: \[ 2x=-10. \] Divide by 2: \[ x=-5. \] **(2)** Solve \[ -9+2x=6-x. \] Add \(x\) to both sides: \[ -9+3x=6. \] Add 9 to both sides: \[ 3x=15. \] Divide by 3: \[ x=5. \] **(3)** Solve \[ -6(1-x)=10-2x. \] Distribute \(-6\): \[ -6+6x=10-2x. \] Add \(2x\) to both sides: \[ -6+8x=10. \] Add 6 to both sides: \[ 8x=16. \] Divide by 8: \[ x=2. \] **(4)** Solve \[ 9(x-3)=5(x-1)-14. \] Distribute on both sides: \[ 9x-27=5x-5-14. \] Simplify the right-hand side: \[ 9x-27=5x-19. \] Subtract \(5x\) from both sides: \[ 4x-27=-19. \] Add 27 to both sides: \[ 4x=8. \] Divide by 4: \[ x=2. \] **(5)** Solve \[ \frac{2}{3}(x+2)-\frac{x+1}{5}=3. \] Multiply every term by the least common denominator (15): \[ 15\cdot\frac{2}{3}(x+2)-15\cdot\frac{x+1}{5}=15\cdot3. \] This simplifies to: \[ 5\cdot2(x+2)-3(x+1)=45. \] \[ 10(x+2)-3(x+1)=45. \] Expand: \[ 10x+20-3x-3=45. \] Combine like terms: \[ 7x+17=45. \] Subtract 17: \[ 7x=28. \] Divide by 7: \[ x=4. \] **(6)** Solve \[ \frac{x-1}{2}-\frac{x+6}{3}=-2. \] Multiply every term by the least common denominator (6): \[ 6\cdot\frac{x-1}{2}-6\cdot\frac{x+6}{3}=6\cdot(-2). \] Simplify: \[ 3(x-1)-2(x+6)=-12. \] \[ 3x-3-2x-12=-12. \] Combine like terms: \[ x-15=-12. \] Add 15 to both sides: \[ x=3. \] **(7)** Solve \[ x(x-2)+1=(x-2)^2. \] Expand both sides: \[ x^2-2x+1=x^2-4x+4. \] Subtract \(x^2\) from both sides: \[ -2x+1=-4x+4. \] Add \(4x\) to both sides: \[ 2x+1=4. \] Subtract 1: \[ 2x=3. \] Divide by 2: \[ x=\frac{3}{2}. \] **(8)** Solve \[ (x+2)(x-2)=(x+3)^2. \] Expand both sides: \[ x^2-4=x^2+6x+9. \] Subtract \(x^2\) from both sides: \[ -4=6x+9. \] Subtract 9 from both sides: \[ -13=6x. \] Divide by 6: \[ x=-\frac{13}{6}. \]