9^(-x+1)+8(3)^(-x) > 1

Question

Solve the inequalities, thank you

Answer 1

 

Hi Babi,

 

9^-x+1 + 8·3^-x > 1

 

9^-x+1 + 8·3^-x -1 > 0

 

(3²)^-x+1 + 8·3^-x -1 > 0

 

3^-2x+2 + 8·3^-x -1 > 0

 

3²·3^-2x + 8·3^-x -1 > 0

 

9·3^-2x + 8·3^-x -1 > 0

 

3^2x - 8·3^x - 9 < 0     (both sides multipled by -32x, and

                               the direction of the inequality reversed)

 

If we let Y ≡ 3^x,

the above inequality becomes

 

Y² - 8Y - 9 < 0

 

(Y+1)(Y-9) < 0

 

-1 < Y < 9

 

-1 < 3^x < 9

 

As 3^x is positive, -1 < 3^x holds for any real x.

 

Now, look at 3^x < 9.

 

3^x < 9

 

3^x < 3²

 

So, x < 2.

 

The final answer is x < 2.