Solve for x
x\in \left(-\infty,-4\right)\cup \left(-\frac{5}{2},\infty\right)
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x+4>0 x+4<0
Denominator x+4 cannot be zero since division by zero is not defined. There are two cases.
x>-4
Consider the case when x+4 is positive. Move 4 to the right hand side.
2-x<3\left(x+4\right)
The initial inequality does not change the direction when multiplied by x+4 for x+4>0.
2-x<3x+12
Multiply out the right hand side.
-x-3x<-2+12
Move the terms containing x to the left hand side and all other terms to the right hand side.
-4x<10
Combine like terms.
x>-\frac{5}{2}
Divide both sides by -4. Since -4 is negative, the inequality direction is changed.
x>-\frac{5}{2}
Consider condition x>-4 specified above. The result remains the same.
x<-4
Now consider the case when x+4 is negative. Move 4 to the right hand side.
2-x>3\left(x+4\right)
The initial inequality changes the direction when multiplied by x+4 for x+4<0.
2-x>3x+12
Multiply out the right hand side.
-x-3x>-2+12
Move the terms containing x to the left hand side and all other terms to the right hand side.
-4x>10
Combine like terms.
x<-\frac{5}{2}
Divide both sides by -4. Since -4 is negative, the inequality direction is changed.
x<-4
Consider condition x<-4 specified above.
x\in \left(-\infty,-4\right)\cup \left(-\frac{5}{2},\infty\right)
The final solution is the union of the obtained solutions.
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