1. We are asked to solve the equation $$3 \sin(2x) + 5x - 4 = 0$$ for $x$. 2. This is a transcendental equation because it contains both the sine function and a linear term in $x$, so we likely cannot solve it algebraically for exact values. 3. Rewrite the equation as $$3 \sin(2x) = 4 - 5x$$. 4. We want to find values of $x$ where the sine function scaled by 3 matches $4 - 5x$. 5. Since $\sin(2x)$ ranges between $-1$ and $1$, the left side ranges between $-3$ and $3$. 6. For solutions, $4 - 5x$ must be within $[-3,3]$. So: - $$-3 \leq 4 - 5x \leq 3$$ - Subtract 4: $$-7 \leq -5x \leq -1$$ - Divide by -5 (flip inequality): $$\frac{7}{5} \geq x \geq \frac{1}{5}$$ 7. So only $x$ in $$\left[\frac{1}{5}, \frac{7}{5}\right]$$ can solve the equation. 8. Next, use numerical methods such as the Newton-Raphson or bisection method in that interval to find approximate roots. 9. For example, at $x=\frac{1}{5}=0.2$: - Left side: $3 \sin(2 \times 0.2) + 5 \times 0.2 - 4 = 3 \sin(0.4) + 1 - 4$. - Calculate $\sin(0.4) \approx 0.389418$, so approx $3 \times 0.389418 + 1 - 4 = 1.168254 + 1 -4 = -1.831746$ (negative). 10. At $x = \frac{7}{5} = 1.4$: - Left side: $3 \sin(2 \times 1.4) + 5 \times 1.4 - 4 = 3 \sin(2.8) + 7 -4$. - Calculate $\sin(2.8) \approx 0.334988$, so approx $3 \times 0.334988 + 3 = 1.004964 + 3 = 4.004964$ (positive). 11. Since the function changes from negative to positive between $0.2$ and $1.4$, a root exists there. 12. Using numerical solver or Newton's method (not shown fully here), the root approximates to about $x \approx 1.27$. 13. Depending on the interval considered, similar methods can be used to find other roots if any. 14. Final solution in approximate form: $$x \approx 1.27$$ within the domain $$\left[0.2, 1.4\right]$$ where the equation holds.