Chapter ISG, Problem 3.8.16P

To determine

To Find: The absolute value equation for each situation and solution.

Answer to Problem 3.8.16P

1. $|x-(2x+2)|=4$

Solution is 2, -6

2. $|2x+2|=x-3$

Solution is -5, $\frac{1}{3}$

3. $-3|x-2|=1+5x$

Solution is $\frac{5}{8},\frac{-7}{2}$

Explanation of Solution

Given:

1. The absolute value of the difference between a number and twice the number plus 2 is 4.
2. The absolute value of the sum of twice a number and 2 is the number minus 3.
3. Negative three times the absolute value of the difference between a number and 2 is the sum of 1 and five times the number.

As per definition of absolute value function:

$\left|x\right|=\{\begin{array}{c}x,x>0\\ -x,x<0\end{array}$

1. Let the number be $x$ .

As per given statement, $|x-(2x+2)|=4$

$|-x-2|=4\Rightarrow |x+2|=4$

Using the above definition of the absolute value function:

$\begin{array}{l}x+2=4\Rightarrow x=2\\ -x-2=4\Rightarrow x=-6\end{array}$

2. Let the number be $x$ .

As per given statement, $|2x+2|=x-3$

Using the above definition of the absolute value function:

$\begin{array}{l}2x+2=x-3\Rightarrow x=-5\\ -2x-2=x-3\Rightarrow x=\frac{1}{3}\end{array}$

3. Let the number be $x$ .

As per given statement, $-3|x-2|=1+5x$

Using the above definition of the absolute value function:

$\begin{array}{l}-3x+6=1+5x\Rightarrow x=\frac{5}{8}\\ 3x-6=1+5x\Rightarrow x=\frac{-7}{2}\end{array}$