Exercise 6 Page 45

How do we solve absolute value equations?

See solution.

Practice makes perfect

Let's start by reviewing how the absolute value function works. The absolute value of a real number |x| is the distance from 0 on the number line. Then, |2|=2 and |-2|=2.

To take both possibilities into account when solving an absolute value equation, we isolate the expression inside the absolute value and then derive 2 equations. One using the negative value, and the other using the positive value. We can see an example below.

2|x+4|=10x

DivEqn

.LHS /2.=.RHS /2.

2|x+4|/2=10x/2

CalcQuot

Calculate quotient

|x+4|=5x

Now, we can derive the positive and the negative case. Derived Equations 0.5cm [0.8em] x+4 = 5x 1cm x+4 =-5x If we are solving an absolute value equation and one of the derived equations solution is not a solution to the original equation, that is called an extraneous solution. Let's solve the derived equations shown above.

x+4 = 5x

SubEqn

LHS-x=RHS-x

x+4-x = 5x-x

SubTerms

Subtract terms

4= 4x

DivEqn

.LHS /4.=.RHS /4.

4/4 = 4x/4

CalcQuot

Calculate quotient

1=x

RearrangeEqn

Rearrange equation

x=1

Since the other equation just differ in the sign of the right-hand side, the solution is x=-1. Now we should check if these solutions make the original equation hold true. Let's first try using x=1

|x+4|= 5x

Substitute

x= 1

| 1+4| ? = 5( 1)

Multiply

|1+4| ? = 5

AddTerms

Add terms

|5|? = 5

AbsPos

|5|=5

5=5