Solve 6p-5/p+6<3

$$\frac { 6 p - 5 } { p + 6 } < 3$$

Solve for p

$$p\in \left(-6,\frac{23}{3}\right)$$

Solution Steps

Denominator p+6 cannot be zero since division by zero is not defined. There are two cases.

$$p+6>0 p+6<0$$

Consider the case when p+6 is positive. Move 6 to the right hand side.

$$p>-6$$

The initial inequality does not change the direction when multiplied by p+6 for p+6>0.

$$6p-5<3\left(p+6\right)$$

Multiply out the right hand side.

$$6p-5<3p+18$$

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

$$6p-3p<5+18$$

Combine like terms.

$$3p<23$$

Divide both sides by 3. Since 3 is positive, the inequality direction remains the same.

$$p<\frac{23}{3}$$

Consider condition p>-6 specified above.

$$p\in \left(-6,\frac{23}{3}\right)$$

Now consider the case when p+6 is negative. Move 6 to the right hand side.

$$p<-6$$

The initial inequality changes the direction when multiplied by p+6 for p+6<0.

$$6p-5>3\left(p+6\right)$$

Multiply out the right hand side.

$$6p-5>3p+18$$

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

$$6p-3p>5+18$$

Combine like terms.

$$3p>23$$

Divide both sides by 3. Since 3 is positive, the inequality direction remains the same.

$$p>\frac{23}{3}$$

Consider condition p<-6 specified above.

$$p\in \emptyset$$

The final solution is the union of the obtained solutions.

$$p\in \left(-6,\frac{23}{3}\right)$$