\( \leqslant \) Solve the following equation by making an appropriate substitution. \[ 3 x^{-2}-13 x^{-1}+12=0 \] The solution set is \( \} \). Tvpe an integer or a simolified fraction Tyne exact answers using radicals as nee

Solve the equation \( 3x^{-2}-13x^{-1}+12=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3x^{-2}-13x^{-1}+12=0\) - step1: Find the domain: \(3x^{-2}-13x^{-1}+12=0,x\neq 0\) - step2: Rewrite the expression: \(\frac{3}{x^{2}}-\frac{13}{x}+12=0\) - step3: Multiply both sides of the equation by LCD: \(\left(\frac{3}{x^{2}}-\frac{13}{x}+12\right)x^{2}=0\times x^{2}\) - step4: Simplify the equation: \(3-13x+12x^{2}=0\) - step5: Factor the expression: \(\left(1-3x\right)\left(3-4x\right)=0\) - step6: Separate into possible cases: \(\begin{align}&1-3x=0\\&3-4x=0\end{align}\) - step7: Solve the equation: \(\begin{align}&x=\frac{1}{3}\\&x=\frac{3}{4}\end{align}\) - step8: Check if the solution is in the defined range: \(\begin{align}&x=\frac{1}{3}\\&x=\frac{3}{4}\end{align},x\neq 0\) - step9: Find the intersection: \(\begin{align}&x=\frac{1}{3}\\&x=\frac{3}{4}\end{align}\) - step10: Rewrite: \(x_{1}=\frac{1}{3},x_{2}=\frac{3}{4}\) The solution set for the equation \(3x^{-2}-13x^{-1}+12=0\) is \(x = \frac{1}{3}\) or \(x = \frac{3}{4}\).