Solving inequalities containing exponential functions, eg, $\frac{(x^2-4)3x}{2^{x-2}(8x+16)}\leq 0$

Question

I know solving inequalities using table method ,i.e , finding each values of variable that makes it zero , and writing the root a line and change the sign when you have odd root. It is classical high school method. However , i encounter with different question style such that it contains exponential function. However , i do not know how to solve them when exponential functions involved in , so i want to learn the techniques for them. For example , how can i solve :

$$\frac{(x^2-4)3x}{2^{x-2}(8x+16)}\leq 0$$

What is the role of exponential function , should i include $(x-2)=0 \rightarrow 2$ or the sign of "x" , it is positive.

NOTE I am open to any other source to read and improve myself about it

https://math.stackexchange.com/questions/317528/how-do-you-compute-negative-numbers-to-fractional-powers and https://math.stackexchange.com/questions/1392545/exponential-functions-with-negative-base @ilikemath — insipidintegrator, Jul 30 '22 at 10:01

score 1 · Answer 1 · answered Jul 30 '22 at 09:15

1

it's equivalent to $x(x-2) \leq 0,$ therefore the set of solutions is $[0,2].$

answered Jul 30 '22 at 09:15

W.Döblin

31

score 0 · Answer 2 · answered Jul 30 '22 at 09:17

0

Simply note that $2^{x-2}>0 \space\forall \space x \in \mathbb R$. The inequality now reduces to $$\frac{3x(x^2-4)}{8x+16}\le0$$ which is something you mentioned as being comfortable with.

answered Jul 30 '22 at 09:17

Cathedral

1,185

Solving inequalities containing exponential functions, eg, $\frac{(x^2-4)3x}{2^{x-2}(8x+16)}\leq 0$

2 Answers2