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

Let's solve each equation step by step! 1. \( 7 + 9x = 7x - 3 \) Subtract \( 7x \) from both sides: \( 9x - 7x = -3 - 7 \) \( 2x = -10 \) Divide by 2: \( x = -5 \) 2. \( -9 + 2x = 6 - x \) Add \( x \) to both sides: \( -9 + 3x = 6 \) Add 9: \( 3x = 15 \) Divide by 3: \( x = 5 \) 3. \( -6(1 - x) = 10 - 2x \) Distribute the -6: \( -6 + 6x = 10 - 2x \) Add \( 2x \): \( -6 + 8x = 10 \) Add 6: \( 8x = 16 \) Divide by 8: \( x = 2 \) 4. \( 9(x - 3) = 5(x - 1) - 14 \) Distribute: \( 9x - 27 = 5x - 5 - 14 \) Combine terms on the right: \( 9x - 27 = 5x - 19 \) Subtract \( 5x \): \( 4x - 27 = -19 \) Add 27: \( 4x = 8 \) Divide by 4: \( x = 2 \) 5. \( \frac{2}{3}(x + 2) - \frac{x + 1}{5} = 3 \) Multiply through by 15 (common denominator): \( 10(x + 2) - 3(x + 1) = 45 \) Expand: \( 10x + 20 - 3x - 3 = 45 \) Combine: \( 7x + 17 = 45 \) Subtract 17: \( 7x = 28 \) Divide by 7: \( x = 4 \) 6. \( \frac{x - 1}{2} - \frac{x + 6}{3} = -2 \) Multiply through by 6 (common denominator): \( 3(x - 1) - 2(x + 6) = -12 \) Expand: \( 3x - 3 - 2x - 12 = -12 \) Combine: \( x - 15 = -12 \) Add 15: \( x = 3 \) 7. \( x(x - 2) + 1 = (x - 2)^2 \) Expand both sides: \( x^2 - 2x + 1 = x^2 - 4x + 4 \) Cancel \( x^2 \): \( -2x + 1 = -4x + 4 \) Add \( 4x \): \( 2x + 1 = 4 \) Subtract 1: \( 2x = 3 \) Divide by 2: \( x = \frac{3}{2} \) 8. \( (x + 2)(x - 2) = (x + 3)^2 \) Expand both sides: \( x^2 - 4 = x^2 + 6x + 9 \) Cancel \( x^2 \): \( -4 = 6x + 9 \) Subtract 9: \( -13 = 6x \) Divide by 6: \( x = -\frac{13}{6} \) The solutions for each equation are: 1. \( x = -5 \) 2. \( x = 5 \) 3. \( x = 2 \) 4. \( x = 2 \) 5. \( x = 4 \) 6. \( x = 3 \) 7. \( x = \frac{3}{2} \) 8. \( x = -\frac{13}{6} \)