Given:
The exponential equation is,
a)
[tex]16^{2x-3}=4^{x+2}[/tex]
Solve the equation,
[tex]\begin{gathered} 16^{2x-3}=4^{x+2} \\ 4^{2(2x-3)}=4^{x+2} \\ \Rightarrow2(2x-3)=x+2 \\ 4x-6=x+2 \\ 4x-x=2+6 \\ 3x=8 \\ x=\frac{8}{3} \end{gathered}[/tex]
Answer: x = 8/3.
b) The inequality is,
[tex]4^{2x+7}\leq32^{2x-3}[/tex]
Solve the inequality,
[tex]\begin{gathered} 4^{2x+7}\leq32^{2x-3} \\ 2^{2(2x+7)}\leq2^{5(2x-3)} \\ \Rightarrow2(2x+7)\leq5(2x-3) \\ 4x+14\leq10x-15 \\ 14+15\leq10x-4x \\ 29\leq6x \\ \frac{29}{6}\leq x \\ x\ge\frac{29}{6} \end{gathered}[/tex]
Answer:
[tex]x\ge\frac{29}{6}[/tex]
