Solve 2x+5/x+1geq3

$$\frac { 2 x + 5 } { x + 1 } \geq 3$$

Solve for x

$$x\in (-1,2]$$

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.

$$2x+5\geq 3\left(x+1\right)$$

Multiply out the right hand side.

$$2x+5\geq 3x+3$$

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

$$2x-3x\geq -5+3$$

Combine like terms.

$$-x\geq -2$$

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

$$x\leq 2$$

Consider condition x>-1 specified above.

$$x\in (-1,2]$$

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.

$$2x+5\leq 3\left(x+1\right)$$

Multiply out the right hand side.

$$2x+5\leq 3x+3$$

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

$$2x-3x\leq -5+3$$

Combine like terms.

$$-x\leq -2$$

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

$$x\geq 2$$

Consider condition x<-1 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in (-1,2]$$