(a) \( \frac{2 x-1}{x+2}=\frac{4}{5} \) (b) \( \sqrt{3} x+\sqrt{12}=\frac{x+5}{\sqrt{2}} \)

Let's solve each equation step by step. ### (a) \( \frac{2x-1}{x+2} = \frac{4}{5} \) To solve this equation, we can cross-multiply: \[ 5(2x - 1) = 4(x + 2) \] Expanding both sides: \[ 10x - 5 = 4x + 8 \] Now, we can isolate \(x\) by moving all terms involving \(x\) to one side and constant terms to the other side: \[ 10x - 4x = 8 + 5 \] This simplifies to: \[ 6x = 13 \] Now, divide both sides by 6: \[ x = \frac{13}{6} \] ### (b) \( \sqrt{3} x + \sqrt{12} = \frac{x + 5}{\sqrt{2}} \) First, let's simplify \(\sqrt{12}\): \[ \sqrt{12} = 2\sqrt{3} \] Now, substitute this back into the equation: \[ \sqrt{3} x + 2\sqrt{3} = \frac{x + 5}{\sqrt{2}} \] Next, we can eliminate the fraction by multiplying both sides by \(\sqrt{2}\): \[ \sqrt{2}(\sqrt{3} x + 2\sqrt{3}) = x + 5 \] Distributing \(\sqrt{2}\): \[ \sqrt{6} x + 2\sqrt{6} = x + 5 \] Now, let's isolate \(x\) by moving all \(x\) terms to one side and constant terms to the other side: \[ \sqrt{6} x - x = 5 - 2\sqrt{6} \] Factoring out \(x\) on the left side: \[ (\sqrt{6} - 1)x = 5 - 2\sqrt{6} \] Now, divide both sides by \((\sqrt{6} - 1)\): \[ x = \frac{5 - 2\sqrt{6}}{\sqrt{6} - 1} \] This is the solution for part (b). If needed, you can rationalize the denominator or simplify further, but this is the exact form of the solution. ### Summary of Solutions: (a) \( x = \frac{13}{6} \) (b) \( x = \frac{5 - 2\sqrt{6}}{\sqrt{6} - 1} \)