Exercise 10 Page 31

Break down the given absolute value equation into two separate equations.

x=6

Practice makes perfect

When the absolute value of an expression is equal to an expression, either the expressions are equal or the opposites of the expressions are equal. Let's look at an example equation. |ax+b|=cx For this equation, there are two possible cases to consider. ax+b=cx or ax+b=- cx To solve the given absolute value equation, we will write two equations similar to those above when we remove the absolute value.

|x+6|=2x

lc x+6 ≥ 0:x+6 = 2x & (I) x+6 < 0:x+6 = - 2x & (II)

lcx+6=2x & (I) x+6=- 2x & (II)

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

l6=x 6=- 3x

DivEqn

(II):.LHS /(- 3).=.RHS /(- 3).

l6=x - 2=x

RearrangeEqn

(II): Rearrange equation

lx= 6 x= - 2

After solving an absolute value equation, it is necessary to check for extraneous solutions. To do this, we substitute the found solutions into the given equation and determine if they make true statements. Let's start with x= 6.

|x+6|=2x

Substitute

x= 6

|{\color{#0000FF}{6}}+6|\stackrel{?}=2( {\color{#0000FF}{6}})

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Simplify

AddTerms

Add terms

|12|\stackrel{?}=2(6)

Multiply

|12|\stackrel{?}=12

AbsPos