Solve x^2/5-x+x>-1

$$\frac{ { x }^{ 2 } }{ 5-x } +x > -1$$

Solve for x

$$x\in \left(-\frac{5}{4},5\right)$$

Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{5-x}{5-x}.

$$\frac{x^{2}}{5-x}+\frac{x\left(5-x\right)}{5-x}>-1$$

Since \frac{x^{2}}{5-x} and \frac{x\left(5-x\right)}{5-x} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+x\left(5-x\right)}{5-x}>-1$$

Do the multiplications in x^{2}+x\left(5-x\right).

$$\frac{x^{2}+5x-x^{2}}{5-x}>-1$$

Combine like terms in x^{2}+5x-x^{2}.

$$\frac{5x}{5-x}>-1$$

Denominator 5-x cannot be zero since division by zero is not defined. There are two cases.

$$5-x>0 5-x<0$$

Consider the case when 5-x is positive. Move 5 to the right hand side.

$$-x>-5$$

Divide both sides by -1. Since -1 is negative, the inequality direction is changed.

$$x<5$$

The initial inequality does not change the direction when multiplied by 5-x for 5-x>0.

$$5x>-\left(5-x\right)$$

Multiply out the right hand side.

$$5x>-5+x$$

Move the terms containing x to the left hand side and all other terms to the right hand side.

$$5x-x>-5$$

Combine like terms.

$$4x>-5$$

Divide both sides by 4. Since 4 is positive, the inequality direction remains the same.

$$x>-\frac{5}{4}$$

Consider condition x<5 specified above.

$$x\in \left(-\frac{5}{4},5\right)$$

Now consider the case when 5-x is negative. Move 5 to the right hand side.

$$-x<-5$$

Divide both sides by -1. Since -1 is negative, the inequality direction is changed.

$$x>5$$

The initial inequality changes the direction when multiplied by 5-x for 5-x<0.

$$5x<-\left(5-x\right)$$

Multiply out the right hand side.

$$5x<-5+x$$

Move the terms containing x to the left hand side and all other terms to the right hand side.

$$5x-x<-5$$

Combine like terms.

$$4x<-5$$

Divide both sides by 4. Since 4 is positive, the inequality direction remains the same.

$$x<-\frac{5}{4}$$

Consider condition x>5 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\frac{5}{4},5\right)$$