How do I solve this inequality? Please include steps

mmm4444bot

Super Moderator
Staff member
Joined
Oct 6, 2005
Messages
10,617

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

🤖
 

HallsofIvy

Elite Member
Joined
Jan 27, 2012
Messages
7,770
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
Joined
Dec 30, 2014
Messages
11,173
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.
 

Otis

Elite Member
Joined
Apr 22, 2015
Messages
3,755
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
Joined
Jan 27, 2012
Messages
7,770
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.
 
Top