Solve f(x)=2x-1/3-x:(-6)<5

$$f ( x ) = \frac { 2 x - 1 } { 3 - x } : ( - 6 ) < 5$$

Solve for x

$$x\in \left(-\infty,3\right)\cup \left(\frac{89}{28},\infty\right)$$

Solution Steps

Multiply both sides by -6. Since -6 is negative, the inequality direction is changed.

$$\frac{2x-1}{3-x}>5\left(-6\right)$$

Multiply 5 and -6 to get -30.

$$\frac{2x-1}{3-x}>-30$$

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

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

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

$$-x>-3$$

Divide both sides by -1. Since -1 is negative, the inequality direction is changed.

$$x<3$$

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

$$2x-1>-30\left(3-x\right)$$

Multiply out the right hand side.

$$2x-1>-90+30x$$

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

$$2x-30x>1-90$$

Combine like terms.

$$-28x>-89$$

Divide both sides by -28. Since -28 is negative, the inequality direction is changed.

$$x<\frac{89}{28}$$

Consider condition x<3 specified above.

$$x<3$$

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

$$-x<-3$$

Divide both sides by -1. Since -1 is negative, the inequality direction is changed.

$$x>3$$

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

$$2x-1<-30\left(3-x\right)$$

Multiply out the right hand side.

$$2x-1<-90+30x$$

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

$$2x-30x<1-90$$

Combine like terms.

$$-28x<-89$$

Divide both sides by -28. Since -28 is negative, the inequality direction is changed.

$$x>\frac{89}{28}$$

Consider condition x>3 specified above. The result remains the same.

$$x>\frac{89}{28}$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\infty,3\right)\cup \left(\frac{89}{28},\infty\right)$$