Solve 10-x/2-x<-11/5

$$\frac { 10 - x } { 2 - x } < \frac { - 11 } { 5 }$$

Solve for x

$$x\in \left(2,\frac{9}{2}\right)$$

Solution Steps

Fraction \frac{-11}{5} can be rewritten as -\frac{11}{5} by extracting the negative sign.

$$\frac{10-x}{2-x}<-\frac{11}{5}$$

Denominator 2-x cannot be zero since division by zero is not defined. There are two cases.

$$2-x>0 2-x<0$$

Consider the case when 2-x is positive. Move 2 to the right hand side.

$$-x>-2$$

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

$$x<2$$

The initial inequality does not change the direction when multiplied by 2-x for 2-x>0.

$$10-x<-\frac{11}{5}\left(2-x\right)$$

Multiply out the right hand side.

$$10-x<-\frac{22}{5}+\frac{11}{5}x$$

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

$$-x-\frac{11}{5}x<-10-\frac{22}{5}$$

Combine like terms.

$$-\frac{16}{5}x<-\frac{72}{5}$$

Divide both sides by -\frac{16}{5}. Since -\frac{16}{5} is negative, the inequality direction is changed.

$$x>\frac{9}{2}$$

Consider condition x<2 specified above.

$$x\in \emptyset$$

Now consider the case when 2-x is negative. Move 2 to the right hand side.

$$-x<-2$$

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

$$x>2$$

The initial inequality changes the direction when multiplied by 2-x for 2-x<0.

$$10-x>-\frac{11}{5}\left(2-x\right)$$

Multiply out the right hand side.

$$10-x>-\frac{22}{5}+\frac{11}{5}x$$

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

$$-x-\frac{11}{5}x>-10-\frac{22}{5}$$

Combine like terms.

$$-\frac{16}{5}x>-\frac{72}{5}$$

Divide both sides by -\frac{16}{5}. Since -\frac{16}{5} is negative, the inequality direction is changed.

$$x<\frac{9}{2}$$

Consider condition x>2 specified above.

$$x\in \left(2,\frac{9}{2}\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(2,\frac{9}{2}\right)$$