Solve for \( x \), if possible (a) \( \sqrt{x-4}=5=0 \) (d) \( \sqrt{3 x-2}-x \equiv 0 \) (3) \( \sqrt{4-2 x}-\sqrt{x+1}=0 \)

Solve the equation \( \sqrt{x-4}=5=0 \). Solve the system of equations by following steps: - step0: Solution: \(\left\{ \begin{array}{l}\sqrt{x-4}=5\\5=0\end{array}\right.\) - step1: Calculate: \(\left\{ \begin{array}{l}x=29\\x \in \varnothing \end{array}\right.\) - step2: Find the intersection: \(x \in \varnothing \) Solve the equation \( \sqrt{3x-2}-x=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sqrt{3x-2}-x=0\) - step1: Find the domain: \(\sqrt{3x-2}-x=0,x\geq \frac{2}{3}\) - step2: Move the variable to the right-hand side: \(\sqrt{3x-2}=x\) - step3: Evaluate: \(\sqrt{3x-2}=x,x\geq 0\) - step4: Solve the equation: \(\begin{align}&x=2\\&x=1\end{align},x\geq 0\) - step5: Find the intersection: \(\begin{align}&x=2\\&x=1\end{align}\) - step6: Check if the solution is in the defined range: \(\begin{align}&x=2\\&x=1\end{align},x\geq \frac{2}{3}\) - step7: Find the intersection: \(\begin{align}&x=2\\&x=1\end{align}\) - step8: Check the solution: \(\begin{align}&x=2\\&x=1\end{align}\) - step9: Rewrite: \(x_{1}=1,x_{2}=2\) Solve the equation \( \sqrt{4-2x}-\sqrt{x+1}=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sqrt{4-2x}-\sqrt{x+1}=0\) - step1: Find the domain: \(\sqrt{4-2x}-\sqrt{x+1}=0,-1\leq x\leq 2\) - step2: Move the expression to the right-hand side: \(\sqrt{4-2x}=\sqrt{x+1}\) - step3: Raise both sides to the \(2\)-th power\(:\) \(\left(\sqrt{4-2x}\right)^{2}=\left(\sqrt{x+1}\right)^{2}\) - step4: Evaluate the power: \(4-2x=x+1\) - step5: Move the expression to the left side: \(-2x-x=1-4\) - step6: Add and subtract: \(-3x=1-4\) - step7: Add and subtract: \(-3x=-3\) - step8: Change the signs: \(3x=3\) - step9: Divide both sides: \(\frac{3x}{3}=\frac{3}{3}\) - step10: Divide the numbers: \(x=1\) - step11: Check if the solution is in the defined range: \(x=1,-1\leq x\leq 2\) - step12: Find the intersection: \(x=1\) - step13: Check the solution: \(x=1\) The solutions to the given equations are: (a) \( \sqrt{x-4}=5=0 \) has no solution. (b) \( \sqrt{3x-2}-x=0 \) has solutions \( x=1 \) and \( x=2 \). (c) \( \sqrt{4-2x}-\sqrt{x+1}=0 \) has solution \( x=1 \).