When solving an equation involving absolute value expressions, we should consider what would happen if we removed the absolute value symbols. Let's look at an example equation.

∣ a x + b ∣ = ∣ c x + d ∣

Although we can make

4

statements about this equation, there are actually only two possible cases to consider.

Statement	Result
Both absolute values are positive.	$a x + b = c x + d$
Both absolute values are negative.	$- (a x + b) = - (c x + d)$
Only the left-hand side is negative.	$- (a x + b) = c x + d$
Only the right-hand side is negative.	$a x + b = - (c x + d)$

Because of the Properties of Equality, when the absolute values of two expressions are equal, either

the

expressions

are

equal

or

the

opposites

of

the

expressions

are

equal .

Now let's consider these two cases for the given equation.

Given Equation : First Equation : Second Equation : ∣ 2 x + 2 ∣ 2 x + 2 2 x + 2 = ∣ x - 2 ∣ = x - 2 = - (x - 2)

We will solve each of these equations by graphing separately.

First Equation

To graph the first equation, we will create two functions out of the left- and right-hand sides of the equation.

y = 2 x + 2 and y = x - 2

The

x -

coordinate where the graphs of these functions intersect is the solution to our equation.

From the graph, we can see that the graphs intersect at

(- 4, - 6)

which means the solution to the equation is

x = - 4 .

Let's check whether it is correct by substituting it into the original equation.

∣ 2 x + 2 ∣ = ∣ x - 2 ∣

Substitute

$x = - 4$

∣ 2 (- 4) + 2 ∣ = ? ∣ - 4 - 2 ∣

▼

Simplify

MultPosNeg

$a (- b) = - a \cdot b$

∣ - 8 + 2 ∣ = ? ∣ - 4 - 2 ∣

AddSubTerms

Add and subtract terms

∣ - 6 ∣ = ? ∣ - 6 ∣

AbsNeg

$∣ - 6 ∣ = 6$

6 = 6 ✓

The equation is true, so

x = - 4

solves the equation.

Second Equation

In order to graph the second equation, we will again create functions out of the left- and right-hand sides of the equation.

y = 2 x + 2 and y = - (x - 2)

Once more, the

x -

coordinate where the graphs of these functions intersect is the solution to our equation.

From the graph, we can see that the lines intersect at

(0, 2),

which means the solution to this equation is

x = 0 .

∣ 2 x + 2 ∣ = ∣ x - 2 ∣

Substitute

$x = 0$

∣ 2 (0) + 2 ∣ = ? ∣ 0 - 2 ∣

▼

Simplify

Multiply

∣ 2 ∣ = ? ∣ - 2 ∣

AbsPos

$∣ 2 ∣ = 2$

2 = ? ∣ - 2 ∣

AbsNeg

$∣ - 2 ∣ = 2$

2 = 2 ✓

Since the equation is correct,

x = 0

is another solution to the equation.

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