Solve for x the inequality: 81^(x-1) - 9^(x+1) > 0

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to solve: 81^(x-1) - 9^(x+1) > 0

81^(x-1) - 9^(x+1) > 0

=> 9^2^(x - 1) - 9^(x + 1) > 0

=> 9^(2x - 2) - 9^(x + 1) > 0

=> 9^(2x - 2) > 9^(x + 1)

As 9 is a positive base

=> 2x - 2 > x + 1

=> x > 3

The solution is  x can take values that lie in (3 , +inf.)

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neela | High School Teacher | (Level 3) Valedictorian

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81^(x-1) -9^(x+1)  > 0.

=> 81^(x-1) > 9^(x+1)

9^2(x-1) > 9^(x+1)

We  compare  the exponents as the bases are same .

2(x-1) > x+1

2x-2 > x+1

2x-x > 3

So  x> 3.

Top Answer

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll the base of the 1st term as power of 9:

9^2*(x-1) - 9^(x+1) > 0

We'll shift to the left, the 2nd term:

9^2*(x-1) > 9^(x+1)

Since the bases are bigger than unit value, the function is increasing and we'll get:

2*(x-1)> (x+1)

We'll remove the brackets:

2x - 2 > x + 1

We'll subtract x both sides:

2x - x - 2 > 1

x - 2 > 1

We'll add 2 both sides:

x > 2 + 1

x > 3

The range of values of x, for the inequality to hold, is (3 , +infinite).

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