How do I solve this inequality? Please include steps

jasongousis72

New member
x10^x/11^x>=(x+1)10^(x+1)/11^(x+1)

mmm4444bot

Super Moderator
Staff member

Please share with us how far you get. Can you say what part you find confusing? Thanks.

HallsofIvy

Elite Member
Those exponentials are a nuisance so my first step would be to divide both sides by $$\displaystyle 10^x$$ and multiply both sides by $$\displaystyle 11^x$$. What do you get when you do that?

Jomo

Elite Member
x*x^10 by definition means x*(x*x*x*x*x*x*x*x*x*x) = x*x*x*x*x*x*x*x*x*x*x = x^11
So x*x^10/x^11 = x^11/x^11 = 1.

Now (x+1)^10*(x+1) = [(x+1)*(x+1)*(x+1)*(x+1)*(x+1)*(x+1)*(x+1)*(x+1)*(x+1)*(x+1)]*(x+1) = (x+1)*(x+1)*(x+1)*(x+1)*(x+1)*(x+1)*(x+1)*(x+1)*(x+1)*(x+1)*(x+1) = (x+1)^11

So, (x+1)^10*(x+1)/(x+1)^11 = (x+1)^11/(x+1)^11 =1

For which values of x are both sides equal to one another? Hint: both sides always equal 1.

HallsofIvy

Elite Member
Uh- What I see is NOT "x^10" but "10^x"!

topsquark

Senior Member
Corner time coming...

-Dan

Otis

Elite Member
The following property of exponents is useful for simplifying a ratio of powers having the same base. (We get two such ratios, after applying the steps suggested in post #3.)

a^n / a^m = a^(n-m)

HallsofIvy

Elite Member
x10^x/11^x>=(x+1)10^(x+1)/11^(x+1)
Divide both sides by 10^x to get
x/11^x>= 10(x+1)/11^(x+1)

Multiply both sides by 11^(x+1) to get
11x>= 10(x+ 1)

Distribute the 10 on the right
11x>= 10x+ 10

Subtract 10x from both sides of the equation
x>= 10.