Solve 0.55leq1+0.05-x/0.08+1.6x

$$0.55 \leq 1 + \frac { 0.05 - x } { 0.08 + 1.6 x }$$

Solve for x

$$x\in (-0.05,\frac{43}{140}]$$

Solution Steps

Swap sides so that all variable terms are on the left hand side. This changes the sign direction.

$$1+\frac{0.05-x}{0.08+1.6x}\geq 0.55$$

Subtract 1 from both sides.

$$\frac{0.05-x}{0.08+1.6x}\geq 0.55-1$$

Subtract 1 from 0.55 to get -0.45.

$$\frac{0.05-x}{0.08+1.6x}\geq -0.45$$

Denominator 0.08+1.6x cannot be zero since division by zero is not defined. There are two cases.

$$0.08+1.6x>0 0.08+1.6x<0$$

Consider the case when 0.08+1.6x is positive. Move 0.08 to the right hand side.

$$1.6x>-0.08$$

Divide both sides by 1.6. Since 1.6 is positive, the inequality direction remains the same.

$$x>-0.05$$

The initial inequality does not change the direction when multiplied by 0.08+1.6x for 0.08+1.6x>0.

$$0.05-x\geq -0.45\left(0.08+1.6x\right)$$

Multiply out the right hand side.

$$0.05-x\geq -0.036-0.72x$$

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

$$-x+0.72x\geq -0.05-0.036$$

Combine like terms.

$$-0.28x\geq -0.086$$

Divide both sides by -0.28. Since -0.28 is negative, the inequality direction is changed.

$$x\leq \frac{43}{140}$$

Consider condition x>-0.05 specified above.

$$x\in (-0.05,\frac{43}{140}]$$

Now consider the case when 0.08+1.6x is negative. Move 0.08 to the right hand side.

$$1.6x<-0.08$$

Divide both sides by 1.6. Since 1.6 is positive, the inequality direction remains the same.

$$x<-0.05$$

The initial inequality changes the direction when multiplied by 0.08+1.6x for 0.08+1.6x<0.

$$0.05-x\leq -0.45\left(0.08+1.6x\right)$$

Multiply out the right hand side.

$$0.05-x\leq -0.036-0.72x$$

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

$$-x+0.72x\leq -0.05-0.036$$

Combine like terms.

$$-0.28x\leq -0.086$$

Divide both sides by -0.28. Since -0.28 is negative, the inequality direction is changed.

$$x\geq \frac{43}{140}$$

Consider condition x<-0.05 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in (-0.05,\frac{43}{140}]$$