Solve by graphing: \( \frac{3 x}{2 x-1}=2 x-3 \) \( \left\{-\frac{7}{20}, \frac{21}{10}\right\} \) \( \left\{-1, \frac{3}{4}\right\} \) \( \left\{\frac{3}{10}\right\} \) \( \left\{\frac{3}{10}, \frac{12}{5}\right\} \)

To solve the equation \( \frac{3x}{2x - 1} = 2x - 3 \) by graphing, we'll follow these steps: 1. **Understand the Equation:** - **Left Side (Function 1):** \( y = \frac{3x}{2x - 1} \) - **Right Side (Function 2):** \( y = 2x - 3 \) By graphing these two functions, the solutions to the equation will be the \( x \)-values where the graphs intersect. 2. **Algebraic Solution (for Verification):** While graphing provides a visual solution, solving algebraically ensures accuracy. \[ \frac{3x}{2x - 1} = 2x - 3 \] **Step-by-Step:** - **Multiply both sides by \( 2x - 1 \) to eliminate the fraction:** \[ 3x = (2x - 3)(2x - 1) \] - **Expand the right side:** \[ 3x = 4x^2 - 8x + 3 \] - **Rearrange to form a quadratic equation:** \[ 4x^2 - 11x + 3 = 0 \] - **Solve using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):** \[ x = \frac{11 \pm \sqrt{121 - 48}}{8} = \frac{11 \pm \sqrt{73}}{8} \] - **Approximate Solutions:** \[ x \approx \frac{11 + 8.544}{8} \approx 2.443 \quad \text{and} \quad x \approx \frac{11 - 8.544}{8} \approx 0.307 \] - **Solutions:** \[ x \approx 0.307 \quad \text{and} \quad x \approx 2.443 \] 3. **Matching with Provided Options:** The approximate solutions \( 0.307 \) and \( 2.443 \) correspond closely to the fractions \( \frac{3}{10} \) and \( \frac{12}{5} \) respectively. **Given Options:** - \( \left\{-\frac{7}{20}, \frac{21}{10}\right\} = \{-0.35, 2.1\} \) - \( \left\{-1, \frac{3}{4}\right\} = \{-1, 0.75\} \) - \( \left\{\frac{3}{10}\right\} = \{0.3\} \) - \( \left\{\frac{3}{10}, \frac{12}{5}\right\} = \{0.3, 2.4\} \) **Conclusion:** The correct solutions are \( \frac{3}{10} \) and \( \frac{12}{5} \). **Answer:** \[ \boxed{\left\{\dfrac{3}{10},\, \dfrac{12}{5}\right\}} \]