Solve 3x+7/x+5>5

$$\frac { 3 x + 7 } { x + 5 } > 5$$

Solve for x

$$x\in \left(-9,-5\right)$$

Solution Steps

Denominator x+5 cannot be zero since division by zero is not defined. There are two cases.

$$x+5>0 x+5<0$$

Consider the case when x+5 is positive. Move 5 to the right hand side.

$$x>-5$$

The initial inequality does not change the direction when multiplied by x+5 for x+5>0.

$$3x+7>5\left(x+5\right)$$

Multiply out the right hand side.

$$3x+7>5x+25$$

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

$$3x-5x>-7+25$$

Combine like terms.

$$-2x>18$$

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

$$x<-9$$

Consider condition x>-5 specified above.

$$x\in \emptyset$$

Now consider the case when x+5 is negative. Move 5 to the right hand side.

$$x<-5$$

The initial inequality changes the direction when multiplied by x+5 for x+5<0.

$$3x+7<5\left(x+5\right)$$

Multiply out the right hand side.

$$3x+7<5x+25$$

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

$$3x-5x<-7+25$$

Combine like terms.

$$-2x<18$$

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

$$x>-9$$

Consider condition x<-5 specified above.

$$x\in \left(-9,-5\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-9,-5\right)$$