Solve x+2/5-x>1

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

Solve for x

$$x\in \left(\frac{3}{2},5\right)$$

Solution Steps

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.

$$x+2>5-x$$

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

$$x+x>-2+5$$

Combine like terms.

$$2x>3$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x>\frac{3}{2}$$

Consider condition x<5 specified above.

$$x\in \left(\frac{3}{2},5\right)$$

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.

$$x+2<5-x$$

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

$$x+x<-2+5$$

Combine like terms.

$$2x<3$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x<\frac{3}{2}$$

Consider condition x>5 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(\frac{3}{2},5\right)$$