Solve for x
x\in \left(0,5\right)
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20x-4x^{2}>0
Use the distributive property to multiply 4x by 5-x.
-20x+4x^{2}<0
Multiply the inequality by -1 to make the coefficient of the highest power in 20x-4x^{2} positive. Since -1 is negative, the inequality direction is changed.
4x\left(x-5\right)<0
Factor out x.
x>0 x-5<0
For the product to be negative, x and x-5 have to be of the opposite signs. Consider the case when x is positive and x-5 is negative.
x\in \left(0,5\right)
The solution satisfying both inequalities is x\in \left(0,5\right).
x-5>0 x<0
Consider the case when x-5 is positive and x is negative.
x\in \emptyset
This is false for any x.
x\in \left(0,5\right)
The final solution is the union of the obtained solutions.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}