Solve 84+x/94+xgeq9/10

$$\frac { 84 + x } { 94 + x } \geq \frac { 9 } { 10 }$$

Solve for x

$$x\in (-\infty,-94)\cup [6,\infty)$$

Solution Steps

Denominator 94+x cannot be zero since division by zero is not defined. There are two cases.

$$94+x>0 94+x<0$$

Consider the case when 94+x is positive. Move 94 to the right hand side.

$$x>-94$$

The initial inequality does not change the direction when multiplied by 94+x for 94+x>0.

$$84+x\geq \frac{9}{10}\left(94+x\right)$$

Multiply out the right hand side.

$$84+x\geq \frac{423}{5}+\frac{9}{10}x$$

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

$$x-\frac{9}{10}x\geq -84+\frac{423}{5}$$

Combine like terms.

$$\frac{1}{10}x\geq \frac{3}{5}$$

Divide both sides by \frac{1}{10}. Since \frac{1}{10} is positive, the inequality direction remains the same.

$$x\geq 6$$

Now consider the case when 94+x is negative. Move 94 to the right hand side.

$$x<-94$$

The initial inequality changes the direction when multiplied by 94+x for 94+x<0.

$$84+x\leq \frac{9}{10}\left(94+x\right)$$

Multiply out the right hand side.

$$84+x\leq \frac{423}{5}+\frac{9}{10}x$$

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

$$x-\frac{9}{10}x\leq -84+\frac{423}{5}$$

Combine like terms.

$$\frac{1}{10}x\leq \frac{3}{5}$$

Divide both sides by \frac{1}{10}. Since \frac{1}{10} is positive, the inequality direction remains the same.

$$x\leq 6$$

Consider condition x<-94 specified above.

$$x<-94$$

The final solution is the union of the obtained solutions.

$$x\in (-\infty,-94)\cup [6,\infty)$$