Solve 3+x/5-xgeq0

$$\frac{ 3+x }{ 5-x } \geq 0$$

Solve for x

$$x\in [-3,5)$$

Solution Steps

For the quotient to be ≥0, x+3 and 5-x have to be both ≤0 or both ≥0, and 5-x cannot be zero. Consider the case when x+3\leq 0 and 5-x is negative.

$$x+3\leq 0 5-x<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x+3\geq 0 and 5-x is positive.

$$x+3\geq 0 5-x>0$$

The solution satisfying both inequalities is x\in \left[-3,5\right).

$$x\in [-3,5)$$

The final solution is the union of the obtained solutions.

$$x\in [-3,5)$$