Exercise 20 Page 227

First, notice the following. 23=|n+10| ⇔ |n+10|=23 An absolute value measures an expression's distance from a midpoint on a number line. |n+10|= 23This equation means that the distance is 23, either in the positive direction or the negative direction. |n+10|= 23 ⇒ ln+10= 23 n+10= - 23 To find the solutions to the absolute value equation, we need to solve both of these cases for n.

| n+10|=23

lc n+10 ≥ 0:n+10 = 23 & (I) n+10 < 0:n+10 = - 23 & (II)

lcn+10=23 & (I) n+10=- 23 & (II)

(I), (II): LHS-10=RHS-10

ln_1=13 n_2=- 33

Both n_1=13 and n_2=- 33 solve the equation.

Checking Our Answers

To check those solutions we will substitute them into the original equation. We will start with n_1=13.

23=|n+10|

Substitute

n= 13

23? =| 13+10|

AddTerms

Add terms

23? =|23|

AbsPos

|23|=23

23=23

We got an identity, therefore the solution is correct. Now, we will substitute n_2=- 33.

23=|n+10|

Substitute

n= - 33

23? =|( - 33)+10|

AddTerms

Add terms

23? =|- 23|

AbsNeg

|-23|=23

23=23

Again, we got an identity, therefore the solution is correct.