Solve for x
x>0
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9x^{2}-12x-\left(3x+1\right)^{2}<-1
Use the distributive property to multiply 3x by 3x-4.
9x^{2}-12x-\left(9x^{2}+6x+1\right)<-1
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+1\right)^{2}.
9x^{2}-12x-9x^{2}-6x-1<-1
To find the opposite of 9x^{2}+6x+1, find the opposite of each term.
-12x-6x-1<-1
Combine 9x^{2} and -9x^{2} to get 0.
-18x-1<-1
Combine -12x and -6x to get -18x.
-18x<-1+1
Add 1 to both sides.
-18x<0
Add -1 and 1 to get 0.
x>0
Product of two numbers is <0 if one is >0 and the other is <0. Since -18<0, x must be >0.
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