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if 5 x +(12) x ≥ 13 x , then the solution set for x is ? the answer : ( – ∞,2]

if 5x+(12)x ≥ 13x, then the solution set for x is ?
 the answer : ( – ∞,2]

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2 Answers

Vikas TU
14149 Points
7 years ago
 
Let suppose x > 0
then,
5^x  > 5^0
5^x > 1 …....................(1)
 
for,
x > 0
12^x > 0 ….............(2)
 
for
x > 0
13^x > 1 …............(3)
 
Adding eqns. (1) and (2)
we get,
5^x + 12^x  > 2
 
Take x less than equal to zero
same eqns we would get but for oppsite sign in all the eqns.
thus, eqns. will remain same
we will have final eqns. as,
5^x + 12^x  greater than equal to 2
 
and 
 
13^x also greater than equal to 1
which will be the intersection of the two inequalities,
therfore
x belongs to [2,infinity)
mycroft holmes
272 Points
7 years ago
Rewrite the inequality as (5/13)x + (12/13)x > =  1.
 
Let sin a = 5/13, cos a = 12/13.
 
Then we can write as sinxa + cosxa > = 1.
 
If x > 2, we have sin x a 2 a and cos x a 2 a. And adding the two we get 
 
sinx a + cos x a
 
If x
we have sin x a > sin2 a and cos x a > cos2 a. And adding the two we get 
 
sinx a + cos x a >1.
 
x=2, of course gives equality.
 
So, the solution set is (-infty, 2]
 
 

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