Solve -3x+2/x-4geq-5

$$\frac{ -3x+2 }{ x-4 } \geq -5$$

Solve for x

$$x\in (-\infty,4)\cup [9,\infty)$$

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.

$$-3x+2\geq -5\left(x-4\right)$$

Multiply out the right hand side.

$$-3x+2\geq -5x+20$$

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

$$-3x+5x\geq -2+20$$

Combine like terms.

$$2x\geq 18$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x\geq 9$$

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.

$$-3x+2\leq -5\left(x-4\right)$$

Multiply out the right hand side.

$$-3x+2\leq -5x+20$$

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

$$-3x+5x\leq -2+20$$

Combine like terms.

$$2x\leq 18$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x\leq 9$$

Consider condition x<4 specified above.

$$x<4$$

The final solution is the union of the obtained solutions.

$$x\in (-\infty,4)\cup [9,\infty)$$