Chapter 1.6, Problem 18PPSE

Solution

To solve the equation $\left|4-z\right|-10=1$ and check the answer.

  $-7\text{ and }15$

Given:

The equation, $\left|4-z\right|-10=1$ .

Concept Used:

• The opposite operation of addition is subtraction and the opposite operation of subtraction is addition.
• The opposite operation of multiplication is division and the opposite operation of division is multiplication.
• Distributive property: $a\left(b+c\right)=a\cdot b+a\cdot c$
• The absolute value property: $\text{if }\left|x\right|=a,\text{then}x=\pm a$

Calculation:

In order to solve the given equation $\left|4-z\right|-10=1$ , use the absolute value property and isolate the variable on one side by performing the basic algebraic operations as shown below

  $\begin{array}{c}\left|4-z\right|-10=1\\ \left|4-z\right|-10+10=1+10\\ \left|4-z\right|=11\\ 4-z=\pm 11\\ 4-4-z=\pm 11-4\\ -z=\pm 11-4\\ z=\mp 11+4\\ z=-11+4\text{and }z=11+4\\ z=-7\text{and }z=15\end{array}$

Thus, the solution of the equation are $-7\text{ and }15$ .

In order to check the solution, substitute back $z=-7\text{ and }z=15$ into the given equation one by one and show that the left side expression is equal to the right-side expression as shown below

  $\begin{array}{c}\left|4-z\right|-10=1\\ \left|4-\left(-7\right)\right|-10=1\\ \left|4+7\right|-10=1\\ \left|11\right|-10=1\\ 11-10=1\\ 1=1\text{True statement}\end{array}$

And,

  $\begin{array}{c}\left|4-z\right|-10=1\\ \left|4-15\right|-10=1\\ \left|-11\right|-10=1\\ 11-10=1\\ 1=1\text{True statement}\end{array}$

Thus, $z=-7\text{ and }z=15$ are the solution of the equation.