Solve 2x+4/x-1geq5

$$\frac { 2 x + 4 } { x - 1 } \geq 5$$

Solve for x

$$x\in (1,3]$$

Solution Steps

Denominator x-1 cannot be zero since division by zero is not defined. There are two cases.

$$x-1>0 x-1<0$$

Consider the case when x-1 is positive. Move -1 to the right hand side.

$$x>1$$

The initial inequality does not change the direction when multiplied by x-1 for x-1>0.

$$2x+4\geq 5\left(x-1\right)$$

Multiply out the right hand side.

$$2x+4\geq 5x-5$$

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

$$2x-5x\geq -4-5$$

Combine like terms.

$$-3x\geq -9$$

Divide both sides by -3. Since -3 is negative, the inequality direction is changed.

$$x\leq 3$$

Consider condition x>1 specified above.

$$x\in (1,3]$$

Now consider the case when x-1 is negative. Move -1 to the right hand side.

$$x<1$$

The initial inequality changes the direction when multiplied by x-1 for x-1<0.

$$2x+4\leq 5\left(x-1\right)$$

Multiply out the right hand side.

$$2x+4\leq 5x-5$$

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

$$2x-5x\leq -4-5$$

Combine like terms.

$$-3x\leq -9$$

Divide both sides by -3. Since -3 is negative, the inequality direction is changed.

$$x\geq 3$$

Consider condition x<1 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in (1,3]$$