We are asked for the similarities and differences of solving the absolute value equation $∣ 2 x + 1 ∣ = ∣ x - 7 ∣$ and the equation $4 x + 3 = - 2 x + 9$ by graphing. To do so, we will first go through the whole process of solving the equation and the absolute value equation. Then, we will find the similarities and differences between these processes.

Equation

To solve the equation

4 x + 3 = - 2 x + 9,

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

y = 4 x + 3 and y = - 2 x + 9

The

x -

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

The graphs intersect at $x = 1,$ which is our solution.

Absolute Value Equation

An absolute value equation in the form

∣ a x + b ∣ = ∣ c x + d ∣

has two related equations.

Equation 1 : a x + b Equation 2 : a x + b = - (c x + d = - (c x + d)

Let's write the two related equations for

∣ 2 x + 1 ∣ = ∣ x - 7 ∣ .

Equation 1 : 2 x + 1 Equation 2 : 2 x + 1 = - (x - 7 = - (x - 7)

This means that for the absolute value equation, we have to solve the two related equations in the same way as we have solved

4 x + 3 = - 2 x + 9 .

Let's create the two functions out of the left- and right-hand sides for

2 x + 1 = x - 7 .

y = 2 x + 1 and y = x - 7

Again, the

x -

coordinate where the graphs of these equations intersect is a solution for the absolute value equation.

The graphs intersect at $x = - 8,$ which is our solution. Now, let's create the two functions out of the left- and right-hand sides for $2 x + 1 = - (x - 7) .$

	$2 x + 1 = - (x - 7)$
First Function	$y = 2 x + 1$
Second Function	$y = - (x - 7) \Leftrightarrow y = - x + 7$

Once more, the $x -$ coordinate where the graphs of these functions intersect is the solution to our equation.

The graphs intersect at $x = 2,$ which is our solution. Therefore, for the absolute value equation, we have two solutions, $x = - 8$ and $x = 2 .$

Similarities and Differences

We can now state the similarities and differences between these two processes.

Similarities: When solving the absolute value equation, we write two equivalent equations and solve them in the same way that we solve $4 x + 3 = - 2 x + 9 .$ That is, by writing and graphing a linear equation for each side of the equation. The $x -$ coordinate where the graph of these equations intersects is the solution to the original equation.
Differences: The absolute value equation has two solutions, one for each system formed by its related equations. Conversely, $4 x + 3 = - 2 x + 9$ has only one solution.

Exercise