\[ (2 x-3)^{3 / 2}-3=122 \] The solution set is

Solve the equation \( (2x-3)^{3/2}-3=122 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\left(2x-3\right)^{\frac{3}{2}}-3=122\) - step1: Find the domain: \(\left(2x-3\right)^{\frac{3}{2}}-3=122,x\geq \frac{3}{2}\) - step2: Move the constant to the right side: \(\left(2x-3\right)^{\frac{3}{2}}=122+3\) - step3: Add the numbers: \(\left(2x-3\right)^{\frac{3}{2}}=125\) - step4: Raise both sides to the \(\frac{2}{3}\)-th power\(:\) \(\left(\left(2x-3\right)^{\frac{3}{2}}\right)^{\frac{2}{3}}=125^{\frac{2}{3}}\) - step5: Evaluate the power: \(2x-3=25\) - step6: Move the constant to the right side: \(2x=25+3\) - step7: Add the numbers: \(2x=28\) - step8: Divide both sides: \(\frac{2x}{2}=\frac{28}{2}\) - step9: Divide the numbers: \(x=14\) - step10: Check if the solution is in the defined range: \(x=14,x\geq \frac{3}{2}\) - step11: Find the intersection: \(x=14\) - step12: Check the solution: \(x=14\) The solution set for the equation \((2x-3)^{3/2}-3=122\) is \(x=14\).