Solve x+3/x+4geq1

$$\frac { x + 3 } { x + 4 } \geq 1$$

Solve for x

$$x<-4$$

Solution Steps

Denominator x+4 cannot be zero since division by zero is not defined. There are two cases.

$$x+4>0 x+4<0$$

Consider the case when x+4 is positive. Move 4 to the right hand side.

$$x>-4$$

The initial inequality does not change the direction when multiplied by x+4 for x+4>0.

$$x+3\geq x+4$$

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

$$x-x\geq -3+4$$

Combine like terms.

$$0\geq 1$$

Consider condition x>-4 specified above.

$$x\in \emptyset$$

Now consider the case when x+4 is negative. Move 4 to the right hand side.

$$x<-4$$

The initial inequality changes the direction when multiplied by x+4 for x+4<0.

$$x+3\leq x+4$$

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

$$x-x\leq -3+4$$

Combine like terms.

$$0\leq 1$$

Consider condition x<-4 specified above.

$$x<-4$$

The final solution is the union of the obtained solutions.

$$x<-4$$