1. **State the problem:** Solve the equation $$3^x = 4x + 5$$ for $x$. 2. **Understand the functions:** The left side is an exponential function $y=3^x$, which grows rapidly as $x$ increases. The right side is a linear function $y=4x+5$, a straight line with slope 4 and y-intercept 5. 3. **Graph behavior:** To find solutions, we look for intersections of $y=3^x$ and $y=4x+5$. 4. **Check the graphs described:** - Graph A: exponential decay and negative slope line — does not match our functions. - Graph B: exponential growth and positive slope line — matches our functions. - Graph C: exponential decay and positive slope line — no. - Graph D: exponential growth and positive slope line — also matches. 5. **Check values to find approximate solutions:** - At $x=0$: $3^0=1$, $4(0)+5=5$, so $1<5$. - At $x=1$: $3^1=3$, $4(1)+5=9$, so $3<9$. - At $x=2$: $3^2=9$, $4(2)+5=13$, so $9<13$. - At $x=3$: $3^3=27$, $4(3)+5=17$, so $27>17$. There is a root between $2$ and $3$. 6. **Check negative values:** - At $x=-1$: $3^{-1} = \frac{1}{3} \approx 0.333$, $4(-1)+5=1$, so $0.333<1$. - At $x=-2$: $3^{-2} = \frac{1}{9} \approx 0.111$, $4(-2)+5=-3$, so $0.111>-3$. There is a root between $-2$ and $-1$. 7. **Use numerical methods (e.g., Newton's method or trial) to approximate roots:** - For root between $-2$ and $-1$: Try $x=-1.5$: $3^{-1.5} \approx 0.192$, $4(-1.5)+5= -6+5 = -1$, so $0.192 > -1$. Try $x=-1.2$: $3^{-1.2} \approx 0.28$, $4(-1.2)+5= -4.8+5=0.2$, so $0.28 > 0.2$. Try $x=-1.1$: $3^{-1.1} \approx 0.31$, $4(-1.1)+5= -4.4+5=0.6$, so $0.31 < 0.6$. So root is near $-1.15$. - For root between $2$ and $3$: Try $x=2.5$: $3^{2.5} \approx 15.59$, $4(2.5)+5=10+5=15$, so $15.59 > 15$. Try $x=2.3$: $3^{2.3} \approx 12.99$, $4(2.3)+5=9.2+5=14.2$, so $12.99 < 14.2$. Try $x=2.4$: $3^{2.4} \approx 14.2$, $4(2.4)+5=9.6+5=14.6$, so $14.2 < 14.6$. Try $x=2.45$: $3^{2.45} \approx 14.9$, $4(2.45)+5=9.8+5=14.8$, so $14.9 > 14.8$. Root near $2.44$. 8. **Final approximate solutions:** $$x \approx -1.15, 2.44$$ 9. **Answer:** The solution set is $$\boxed{-1.15, 2.44}$$. 10. **Graph choice:** The correct graph shows exponential growth and a positive slope line intersecting twice, which matches Graph D.