Question

Solve for the values of x for each of the following exponential equations and inequalities.

Answer 1

Given:

The exponential equation is,

a)

`16^(2x-3)=4^(x+2)`

Solve the equation,

`\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=(8)/(3) \end{gathered}`

Answer: x = 8/3.

b) The inequality is,

`4^(2x+7)\leq32^(2x-3)`

Solve the inequality,

`\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 \\ (29)/(6)\leq x \\ x\ge(29)/(6) \end{gathered}`

Answer:

`x\ge(29)/(6)`