Solve 1-x/3-xgeq0

$$\frac { 1 - x } { 3 - x } \geq 0$$

Solve for x

$$x\in (-\infty,1]\cup (3,\infty)$$

Solution Steps

For the quotient to be ≥0, 1-x and 3-x have to be both ≤0 or both ≥0, and 3-x cannot be zero. Consider the case when 1-x\leq 0 and 3-x is negative.

$$1-x\leq 0 3-x<0$$

The solution satisfying both inequalities is x>3.

$$x>3$$

Consider the case when 1-x\geq 0 and 3-x is positive.

$$1-x\geq 0 3-x>0$$

The solution satisfying both inequalities is x\leq 1.

$$x\leq 1$$

The final solution is the union of the obtained solutions.

$$x>3\text{; }x\leq 1$$