Solve 2x+3/x-3geq1

$$\frac { 2 x + 3 } { x - 3 } \geq 1$$

Solve for x

$$x\in (-\infty,-6]\cup (3,\infty)$$

Solution Steps

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

$$x-3>0 x-3<0$$

Consider the case when x-3 is positive. Move -3 to the right hand side.

$$x>3$$

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

$$2x+3\geq x-3$$

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

$$2x-x\geq -3-3$$

Combine like terms.

$$x\geq -6$$

Consider condition x>3 specified above.

$$x>3$$

Now consider the case when x-3 is negative. Move -3 to the right hand side.

$$x<3$$

The initial inequality changes the direction when multiplied by x-3 for x-3<0.

$$2x+3\leq x-3$$

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

$$2x-x\leq -3-3$$

Combine like terms.

$$x\leq -6$$

The final solution is the union of the obtained solutions.

$$x\in (-\infty,-6]\cup (3,\infty)$$