Why does the same inequality give different answers? (log_2(x)−2)(log_2(x)+1)<0 has a solution 1.2<x<4 But when we take the second part alone that is (log_2(x)+1)<0 it gives a solution 0<x<1/2 why is x>1/2 in the first case but x<1/2 in the second case?

Brylee Shepard 2022-08-12 Answered
Why does the same inequality give different answers?
( log 2 ( x ) 2 ) ( log 2 ( x ) + 1 ) < 0
has a solution 1 2 < x < 4
But when we take the second part alone that is
( log 2 ( x ) + 1 ) < 0
it gives a solution 0 < x < 1 2 why is x > 1 2 in the first case but x < 1 2 in the second case?
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Answers (1)

Cindy Walls
Answered 2022-08-13 Author has 10 answers
To solve this, you need exactly one of log 2 ( x ) 2 and log 2 ( x ) + 1 to be negative. The problem reduces to solving the simultaneous inequalities:
log 2 ( x ) 2 < 0  and  log 2 ( x ) + 1 > 0
and
log 2 ( x ) 2 > 0  and  log 2 ( x ) + 1 < 0
to get two solution sets.
So to answer your question, you're correct that log 2 ( x ) + 1 < 0 when 0 < x < 1 2 , but in this region, we also have log 2 ( x ) 2 < 0, so overall the inequality is positive.

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