Homework Help

Solve inequality 2^x+2^(x+2)<=20?

user profile pic

nurli | (Level 2) Honors

Posted July 11, 2013 at 4:44 PM via web

dislike 1 like

Solve inequality 2^x+2^(x+2)<=20?

1 Answer | Add Yours

user profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted July 11, 2013 at 5:12 PM (Answer #1)

dislike 1 like

You need to use the laws of exponents, such that:

`a^(x+y) = a^x*a^y`

Reasoning by analogy, yields:

`2^(x+2) = 2^x*2^2`

Replacing `2^x*2^2` for `2^(x+2)` in inequality, yields:

`2^x + 2^x*2^2 <= 20`

Factoring out `2^x` yields:

`2^x(1 + 4) <= 20 => 2^x*5 <= 20`

You need to divide by 5 both sides, such that:

`2^x <= 4 => 2^x <= 2^2`

Since the base of exponential function is larger than 1, the exponential function increases over its range, hence, the direction of inequality is preserved, such that:

`2^x <= 2^2 => x <= 2 => x in (-oo,2]`

Hence, evaluating the solution to the given inequality, using the laws of exponents and the properties of exponential functions, yields `x in (-oo,2].`

Sources:

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes