Solve 3x-5/2-x<1

$$\frac { 3 x - 5 } { 2 - x } < 1$$

Solve for x

$$x\in \left(-\infty,\frac{7}{4}\right)\cup \left(2,\infty\right)$$

Solution Steps

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.

$$3x-5<2-x$$

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

$$3x+x<5+2$$

Combine like terms.

$$4x<7$$

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

$$x<\frac{7}{4}$$

Consider condition x<2 specified above. The result remains the same.

$$x<\frac{7}{4}$$

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.

$$3x-5>2-x$$

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

$$3x+x>5+2$$

Combine like terms.

$$4x>7$$

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

$$x>\frac{7}{4}$$

Consider condition x>2 specified above.

$$x>2$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\infty,\frac{7}{4}\right)\cup \left(2,\infty\right)$$