1. Solve \(3(2x - 1) - 2(x + 4) = 6\). Distribute: \(6x - 3 - 2x - 8 = 6\). Combine like terms: \(4x - 11 = 6\). Add 11 to both sides: \(4x = 17\). Divide by 4: \(x = \frac{17}{4}\). 2. Solve \(2(1 - 2x) - (x + 3) = 5(1 - x)\). Distribute: \(2 - 4x - x - 3 = 5 - 5x\). Simplify: \(-5x -1 = 5 - 5x\). Add \(5x\) to both sides: \(-1 = 5\) (false), but check again: Rearranging gives \(-5x -1 = 5 - 5x\), add \(5x\) to both sides: \(-1 = 5\), no solution. But recheck: maybe mistake in simplification. Step by step: LHS: \(2 -4x - x - 3 = -5x -1\). RHS: \(5 - 5x\). Set equal: \(-5x -1 = 5 - 5x\). Add \(5x\): \(-1 = 5\), contradiction. So no solution: choose (d) none. 3. Solve \(\frac{3x - 5}{1 - 2x} = \frac{2}{3}\). Cross-multiply: \(3(3x - 5) = 2(1 - 2x)\). Expand: \(9x - 15 = 2 - 4x\). Add \(4x\): \(13x - 15 = 2\). Add 15: \(13x = 17\). Divide by 13: \(x = \frac{17}{13}\). 4. Solve \(\frac{3}{x - 1} + 5 = \frac{7}{x - 1}\). Subtract \(\frac{3}{x - 1}\): \(5 = \frac{7}{x - 1} - \frac{3}{x - 1} = \frac{4}{x - 1}\). Multiply both sides by \(x-1\): \(5(x - 1) = 4\). Expand: \(5x - 5 = 4\). Add 5: \(5x = 9\). Divide: \(x = \frac{9}{5}\). 5. Solve \(\frac{2x - 5}{3x + 7} = \frac{2x + 3}{3x}\). Cross-multiply: \((2x - 5)(3x) = (2x + 3)(3x + 7)\). Expand left: \(6x^2 - 15x\). Expand right: \(6x^2 + 14x + 9x + 21 = 6x^2 + 23x + 21\). Subtract \(6x^2\) from both sides: \(-15x = 23x + 21\). Add \(15x\): \(0 = 38x + 21\). Subtract 21: \(-21 = 38x\). Divide: \(x = -\frac{21}{38}\). 6. Solve \(\frac{3}{x-2} + \frac{4}{x+2} = \frac{1}{x^2 -4}\). Note \(x^2 -4 = (x-2)(x+2)\). Multiply both sides by \((x-2)(x+2)\): \(3(x+2) + 4(x-2) = 1\). Expand: \(3x + 6 + 4x - 8 = 1\). Combine: \(7x - 2 = 1\). Add 2: \(7x = 3\). Divide: \(x = \frac{3}{7}\). 7. Solve \(\frac{2x}{x-3} - 2 = \frac{3}{x+3}\). Multiply both sides by \((x-3)(x+3)\): \(2x(x+3) - 2(x-3)(x+3) = 3(x-3)\). Expand: \(2x^2 + 6x - 2(x^2 - 9) = 3x - 9\). Distribute: \(2x^2 + 6x - 2x^2 + 18 = 3x - 9\). Simplify: \(6x + 18 = 3x - 9\). Subtract \(3x\): \(3x + 18 = -9\). Subtract 18: \(3x = -27\). Divide: \(x = -9\). 8. Solve roots of \(4x^2 + 9x - 17 = 0\). Use quadratic formula: \(x = \frac{-9 \pm \sqrt{9^2 - 4 \times 4 \times (-17)}}{2 \times 4}\). Calculate discriminant: \(81 + 272 = 353\). Roots: \(x = \frac{-9 \pm \sqrt{353}}{8}\). Approximate \(\sqrt{353} \approx 18.79\). First root: \(\frac{-9 + 18.79}{8} \approx 1.223\). Second root: \(\frac{-9 - 18.79}{8} \approx -3.473\). 9. Solve \(6x^2 - 7x + 2 = 0\). Quadratic formula: \(x = \frac{7 \pm \sqrt{(-7)^2 - 4 \times 6 \times 2}}{2 \times 6}\). Discriminant: \(49 - 48 = 1\). Roots: \(x = \frac{7 \pm 1}{12}\). First root: \(\frac{8}{12} = 0.67\). Second root: \(\frac{6}{12} = 0.5\). 10. Solve the system: \(3x - y = -2\) \(x - 3y = 10\) Multiply second by 3: \(3x - 9y = 30\). Subtract first: \((3x - 9y) - (3x - y) = 30 - (-2)\). Simplify: \(-8y = 32\). Divide: \(y = -4\). Substitute into first: \(3x - (-4) = -2\). Simplify: \(3x + 4 = -2\), \(3x = -6\), \(x = -2\). Sum: \(x + y = -2 + (-4) = -6\). **Final Answers:** 1. x = \(\frac{17}{4}\) 2. No solution 3. x = \(\frac{17}{13}\) 4. x = \(\frac{9}{5}\) 5. x = \(-\frac{21}{38}\) 6. x = \(\frac{3}{7}\) 7. x = \(-9\) 8. x = approximately 1.223 or -3.473 9. x = 0.67 or 0.5 10. x + y = -6