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

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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].`


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