How do you solve: [ (x-2) / (x+3) ] < [ (x+1) / (x) ]?
1 Answer
Explanation:
The inequality given to you looks like this
#(x-2)/(x+3) < (x+1)/x#
Right from the start, you know that any solution interval must not contain the values of
More specifically, you need to have
#x+3 != 0 implies x != -3" "# and#" "x != 0#
With that in mind, use the common denominator of the two fractions, which is equal to
More specifically, multiply the first fraction by
This will get you
#(x-2)/(x+3) * x/x < (x+1)/x * (x+3)/(x+3)#
#(x(x-2))/(x(x+3)) < ((x+1)(x+3))/(x(x+3))#
This is equivalent to
#x(x-2) < (x+1)(x+3)#
Expand the parantheses to get
#color(red)(cancel(color(black)(x^2))) - 2x < color(red)(cancel(color(black)(x^2))) + x + 3x + 3#
Rearrange the inequality to isolate
#-6x < 3 implies x > 3/((-6)) <=> x > -1/2#
This means that any value of
Therefore, the solution interval will be