Solve 2x+1/x-2geq3

$$\frac{ 2x+1 }{ x-2 } \geq 3$$

Solve for x

$$x\in (2,7]$$

Solution Steps

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

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

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

$$x>2$$

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

$$2x+1\geq 3\left(x-2\right)$$

Multiply out the right hand side.

$$2x+1\geq 3x-6$$

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

$$2x-3x\geq -1-6$$

Combine like terms.

$$-x\geq -7$$

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

$$x\leq 7$$

Consider condition x>2 specified above.

$$x\in (2,7]$$

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

$$x<2$$

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

$$2x+1\leq 3\left(x-2\right)$$

Multiply out the right hand side.

$$2x+1\leq 3x-6$$

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

$$2x-3x\leq -1-6$$

Combine like terms.

$$-x\leq -7$$

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

$$x\geq 7$$

Consider condition x<2 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in (2,7]$$