How do you solve these 2 inequalities? (Exponential function)

Ok so I'd like an explanation (for one or both) with the answer as well for these 2 problems, please.

1) #(1/4)^x# < #1/8#

2) #(1/4)^x# > #1/8#

Thanks so much!

Answer 1

see below

reduce the fractions to powers of 2

that is # (1/2^2)^x < 1/2^3#

=

#1/2^(2x) < 1/2^3#

now just use the reciprocal and negate the powers

therefore,

#2^(-2x) < 2^-3# therefore,
#-2x < -3#
now when you revert these back to positive, the #<# sign changes to the #># sign
therefore 1) #x > 3/2# = this is the answer for the first question and the second question is just the same whose answer is
2) #x < 3/2#
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Answer 2

To solve inequalities involving exponential functions, follow these general steps:

  1. Isolate the exponential term on one side of the inequality.
  2. Take the natural logarithm (ln) of both sides to eliminate the exponential.
  3. Solve the resulting inequality for the variable.
  4. Check if the solution is valid by ensuring that the original inequality holds true for the obtained solution.
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Answer 3

Sure, I'd be happy to help with exponential inequalities. Could you please provide the specific inequalities you'd like to solve?

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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