Solve a/a+3<1/2

$$\frac { a } { a + 3 } < \frac { 1 } { 2 }$$

Solve for a

$$a\in \left(-3,3\right)$$

Solution Steps

Denominator a+3 cannot be zero since division by zero is not defined. There are two cases.

$$a+3>0 a+3<0$$

Consider the case when a+3 is positive. Move 3 to the right hand side.

$$a>-3$$

The initial inequality does not change the direction when multiplied by a+3 for a+3>0.

$$a<\frac{1}{2}\left(a+3\right)$$

Multiply out the right hand side.

$$a<\frac{1}{2}a+\frac{3}{2}$$

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

$$a-\frac{1}{2}a<\frac{3}{2}$$

Combine like terms.

$$\frac{1}{2}a<\frac{3}{2}$$

Divide both sides by \frac{1}{2}. Since \frac{1}{2} is positive, the inequality direction remains the same.

$$a<3$$

Consider condition a>-3 specified above.

$$a\in \left(-3,3\right)$$

Now consider the case when a+3 is negative. Move 3 to the right hand side.

$$a<-3$$

The initial inequality changes the direction when multiplied by a+3 for a+3<0.

$$a>\frac{1}{2}\left(a+3\right)$$

Multiply out the right hand side.

$$a>\frac{1}{2}a+\frac{3}{2}$$

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

$$a-\frac{1}{2}a>\frac{3}{2}$$

Combine like terms.

$$\frac{1}{2}a>\frac{3}{2}$$

Divide both sides by \frac{1}{2}. Since \frac{1}{2} is positive, the inequality direction remains the same.

$$a>3$$

Consider condition a<-3 specified above.

$$a\in \emptyset$$

The final solution is the union of the obtained solutions.

$$a\in \left(-3,3\right)$$