Solve x+2/x-7geq2/5

$$\frac { x + 2 } { x - 7 } \geq \frac { 2 } { 5 }$$

Solve for x

$$x\in (-\infty,-8]\cup (7,\infty)$$

Solution Steps

Denominator x-7 cannot be zero since division by zero is not defined. There are two cases.

$$x-7>0 x-7<0$$

Consider the case when x-7 is positive. Move -7 to the right hand side.

$$x>7$$

The initial inequality does not change the direction when multiplied by x-7 for x-7>0.

$$x+2\geq \frac{2}{5}\left(x-7\right)$$

Multiply out the right hand side.

$$x+2\geq \frac{2}{5}x-\frac{14}{5}$$

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

$$x-\frac{2}{5}x\geq -2-\frac{14}{5}$$

Combine like terms.

$$\frac{3}{5}x\geq -\frac{24}{5}$$

Divide both sides by \frac{3}{5}. Since \frac{3}{5} is positive, the inequality direction remains the same.

$$x\geq -8$$

Consider condition x>7 specified above.

$$x>7$$

Now consider the case when x-7 is negative. Move -7 to the right hand side.

$$x<7$$

The initial inequality changes the direction when multiplied by x-7 for x-7<0.

$$x+2\leq \frac{2}{5}\left(x-7\right)$$

Multiply out the right hand side.

$$x+2\leq \frac{2}{5}x-\frac{14}{5}$$

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

$$x-\frac{2}{5}x\leq -2-\frac{14}{5}$$

Combine like terms.

$$\frac{3}{5}x\leq -\frac{24}{5}$$

Divide both sides by \frac{3}{5}. Since \frac{3}{5} is positive, the inequality direction remains the same.

$$x\leq -8$$

The final solution is the union of the obtained solutions.

$$x\in (-\infty,-8]\cup (7,\infty)$$