1. **State the problem:** Solve the equation $$3x^4 - x = x^3 + 3$$ by graphing or algebraic manipulation. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$3x^4 - x - x^3 - 3 = 0$$ 3. **Simplify the expression:** $$3x^4 - x^3 - x - 3 = 0$$ 4. **Understand the problem:** We want to find values of $x$ where the quartic polynomial equals the cubic polynomial plus 3. Graphically, these are the $x$-coordinates where the curves intersect. 5. **Use graphing or numerical methods:** The problem suggests solutions near $x=1.09$ and $x=1.25$. 6. **Verify solutions by substitution:** - For $x=1.09$: $$LHS = 3(1.09)^4 - 1.09 \approx 3(1.4095) - 1.09 = 4.2285 - 1.09 = 3.1385$$ $$RHS = (1.09)^3 + 3 \approx 1.295 + 3 = 4.295$$ Values are close but not equal, so $x=1.09$ is an approximate root. - For $x=1.25$: $$LHS = 3(1.25)^4 - 1.25 = 3(2.4414) - 1.25 = 7.3242 - 1.25 = 6.0742$$ $$RHS = (1.25)^3 + 3 = 1.9531 + 3 = 4.9531$$ Again, close but not exact, indicating approximate roots. 7. **Conclusion:** The solutions to the equation are approximately $x \approx 1.09$ and $x \approx 1.25$. **Final answer:** $$x \approx 1.09, 1.25$$