1. **State the problem:** Solve the equation $$6^x = 7x + 4$$ for $x$. 2. **Understand the equation:** This is a transcendental equation where an exponential function equals a linear function. 3. **Use the table to approximate:** The table gives values of $6^x$ and $7x + 4$ for various $x$: | $x$ | $6^x$ | $7x + 4$ | |-----|-------|----------| | 0 | 1 | 4 | | 1 | 6 | 11 | | 1.4 | 12.3 | 13.8 | | 1.45| ? | ? | | 1.5 | 14.7 | 14.5 | | 2 | 36 | 18 | 4. **Check values near $x=1.45$:** - At $x=1.4$, $6^x = 12.3$ and $7x+4=13.8$, so $6^x < 7x+4$. - At $x=1.5$, $6^x = 14.7$ and $7x+4=14.5$, so $6^x > 7x+4$. 5. **Estimate $6^{1.45}$:** Using linear interpolation between $x=1.4$ and $x=1.5$ for $6^x$: $$6^{1.45} \approx 12.3 + \frac{1.45 - 1.4}{1.5 - 1.4} \times (14.7 - 12.3) = 12.3 + 0.5 \times 2.4 = 12.3 + 1.2 = 13.5$$ 6. **Estimate $7(1.45) + 4$:** $$7 \times 1.45 + 4 = 10.15 + 4 = 14.15$$ 7. **Compare at $x=1.45$:** $$6^{1.45} = 13.5 < 14.15 = 7(1.45) + 4$$ 8. **Conclusion:** Since $6^x$ is less than $7x+4$ at $x=1.45$ and greater at $x=1.5$, the solution lies between $1.45$ and $1.5$. 9. **Refine estimate:** Because at $1.5$, $6^x$ is slightly greater, the solution is approximately $x \approx 1.48$. **Final answer:** $$x \approx 1.48$$