Solve x-1/x+2leq3

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

Solve for x

$$x\in (-\infty,-\frac{7}{2}]\cup (-2,\infty)$$

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.

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

Multiply out the right hand side.

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

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

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

Combine like terms.

$$-2x\leq 7$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$x\geq -\frac{7}{2}$$

Consider condition x>-2 specified above.

$$x>-2$$

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.

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

Multiply out the right hand side.

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

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

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

Combine like terms.

$$-2x\geq 7$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$x\leq -\frac{7}{2}$$

The final solution is the union of the obtained solutions.

$$x\in (-\infty,-\frac{7}{2}]\cup (-2,\infty)$$