Solve 350,000+x/80,000+xgeq2.5

$$\frac { 350,000 + x } { 80,000 + x } \geq 2.5$$

Solve for x

$$x\in (-80000,100000]$$

Solution Steps

Denominator 80000+x cannot be zero since division by zero is not defined. There are two cases.

$$80000+x>0 80000+x<0$$

Consider the case when 80000+x is positive. Move 80000 to the right hand side.

$$x>-80000$$

The initial inequality does not change the direction when multiplied by 80000+x for 80000+x>0.

$$350000+x\geq 2.5\left(80000+x\right)$$

Multiply out the right hand side.

$$350000+x\geq 200000+2.5x$$

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

$$x-2.5x\geq -350000+200000$$

Combine like terms.

$$-1.5x\geq -150000$$

Divide both sides by -1.5. Since -1.5 is negative, the inequality direction is changed.

$$x\leq 100000$$

Consider condition x>-80000 specified above.

$$x\in (-80000,100000]$$

Now consider the case when 80000+x is negative. Move 80000 to the right hand side.

$$x<-80000$$

The initial inequality changes the direction when multiplied by 80000+x for 80000+x<0.

$$350000+x\leq 2.5\left(80000+x\right)$$

Multiply out the right hand side.

$$350000+x\leq 200000+2.5x$$

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

$$x-2.5x\leq -350000+200000$$

Combine like terms.

$$-1.5x\leq -150000$$

Divide both sides by -1.5. Since -1.5 is negative, the inequality direction is changed.

$$x\geq 100000$$

Consider condition x<-80000 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in (-80000,100000]$$