Chapter 1.4, Problem 51HP

To determine

To explain: the solving of an absolute value equation of the given form.

Answer to Problem 51HP

  $\left|2x-3\right|=-7x+5$

Explanation of Solution

Given information:

Given equation

  $a\left|x-b\right|+c=d$ for $x$

Concept used:

Absolute value of a function a is given by

  $\left|a\right|=\{\begin{array}{l}aa>0\\ -aa<0\\ 0a=0\end{array}$ \

And

  $\begin{array}{l}\left|-a\right|=a\\ \left|a\right|=a\end{array}$

More over

  $\begin{array}{l}\left|x+a\right|=b\\ \Rightarrow x+a=b\text{or}x+a=-b\end{array}$

Calculation:

Consider the an absolute value equation of the form $a\left|x-b\right|+c=d$ .

  $3\left|2x-3\right|-5=4$

Now, solve the equation as shown,

  $\begin{array}{c}3\left|2x-3\right|-5=4\\ 6x-9-5=4\text{or}\text{}6x-9-5=-4\\ 6x-14=4\text{or}\text{}6x-14=-4\\ 6x-14+14=4+14\text{or}\text{}6x-14+14=-4+14\\ 6x=18\text{or}\text{}6x=10\\ x=3\text{or}x=\frac{5}{3}\end{array}$

Now, check the result by back substitution.

  $\begin{array}{c}3\left|2x-3\right|-5=4\\ 3\left|2\left(3\right)-3\right|-5=4\\ 3\left|3\right|-5=4\\ 9-5=4\\ 4=4\end{array}$

And

  $\begin{array}{c}3\left|2\cdot \frac{5}{3}-3\right|-5=4\\ 3\left|\frac{10}{3}-3\right|-5=4\\ 3\left|\frac{1}{3}\right|-5=4\\ 9-5=4\\ 4=4\end{array}$

Which is true.

Hence, the value of $x$ in the given equation is $x=3\text{or}x=\frac{5}{3}$ .