Solve x+4/x-2leq-5

$$\frac { x + 4 } { x - 2 } \leq - 5$$

Solve for x

$$x\in [1,2)$$

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.

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

Multiply out the right hand side.

$$x+4\leq -5x+10$$

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

$$x+5x\leq -4+10$$

Combine like terms.

$$6x\leq 6$$

Divide both sides by 6. Since 6 is positive, the inequality direction remains the same.

$$x\leq 1$$

Consider condition x>2 specified above.

$$x\in \emptyset$$

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.

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

Multiply out the right hand side.

$$x+4\geq -5x+10$$

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

$$x+5x\geq -4+10$$

Combine like terms.

$$6x\geq 6$$

Divide both sides by 6. Since 6 is positive, the inequality direction remains the same.

$$x\geq 1$$

Consider condition x<2 specified above.

$$x\in [1,2)$$

The final solution is the union of the obtained solutions.

$$x\in [1,2)$$