Chapter 8.6, Problem 1.10A

Solution

To Combine: The steps which used in order to solve inequality $\frac{x}{4-x}<3$ and the case where $4-x<0,\text{ or }x>4$ .

The solution of the is $x>4$ .

Given information:

The given inequality is $\frac{x}{4-x}<3$ .

Explanation:

Consider the given inequality.

  $\frac{x}{4-x}<3$

Multiply the inequality by $4-x$ . Reverse the inequality sign as $4-x<0,\text{ or }x>4$

  $\begin{array}{c}\frac{x}{4-x}\left(4-x\right)>3\left(4-x\right)\\ x>3\left(4-x\right)\end{array}$

Use distributive property.

  $\begin{array}{c}x>3\cdot 4-3\cdot x\\ x>12-3x\end{array}$

Add 3x on both the sides.

  $\begin{array}{c}x+3x>12-3x+3x\\ 4x>12\end{array}$

Divide the both sides by 4.

  $\begin{array}{c}\frac{4x}{4}>\frac{12}{4}\\ x>3\end{array}$

Here it is obtained that $x>3$ but in the case of $4-x<0,\text{ or }x>4$ .

Thus, the combined solution should be $x>4$