In this explainer, we will learn how to solve equations involving the absolute value.

Recall that the **absolute value** of a real number is its distance from 0 on the number line.
For example, in the expression (which can be read as the
absolute value of ), the number is given using
absolute value notation, the two vertical bars. Since is located
5 units from 0 on the number line, the value of the expression is 5. The value of the
expression (which can be read as the absolute value of 5) is also 5, because 5 is
also located 5 units from 0 on the number line.

### Definition: The Absolute Value

The absolute value of a number is the magnitude of the number without regard to its sign:

Let us also recap how to plot the graph of an absolute value function. To do this, we can complete a table of values for :

0 | 1 | 2 | 3 | ||||

3 | 2 | 1 | 0 | 1 | 2 | 3 |

Then, we can plot the coordinates on the Cartesian plane to draw the graph:

Being able to apply the definition of the absolute value and plot absolute value graphs is very useful when solving absolute value equations, so practicing these skills is very important.

Let us now introduce the concept of absolute value equations.

Consider the equation .

Before we look at this algebraically, it can often be useful to think about the problem graphically. So, on the same set of axes, let us plot and . To do this, we can complete a table of values for :

0 | 1 | 2 | 3 | ||||

2 | 1 | 0 | 1 | 2 | 3 | 4 |

From this graph, we can see that the two lines intersect at two distinct points, when and when .

So, the solution to the equation is or . This can be written as a solution set in the form .

Now, we can also see from the graph how we might approach solving this equation algebraically. Recalling our definition of the absolute value function, we can represent this using piecewise notation:

As , we will solve the two equations and .

We start with

Then, we look at

Combining these two equations, we can see that the solutions are or , which agrees with our initial findings from inspecting the graphs.

An alternative algebraic method, as opposed to using piecewise notation, is to set the absolute value expression equal to the positive and the negative of the other quantity.

In this example, , we need to solve two equations: and

Subtracting 1 from both sides of both equations, we once again obtain the solutions and .

This method can be summarized as follows.

### How To: Solving Absolute Value Equations

- Isolate the absolute value expression.
- Set the quantity inside the absolute value notation equal to the positive and the negative of the quantity on the other side of the equation.
- Solve for the unknown in both equations.
- Check the answer analytically or graphically.

The two methods are equivalent, and while we can use either to solve absolute value equations, one or the other of the methods can make the process more efficient.

We will now solve a variety of absolute value equations both algebraically and graphically.

### Example 1: Solving Absolute Value Equations

What is the solution set of the equation ?

### Answer

We will solve this question first by using an algebraic approach and then by using a graphical approach.

To solve this equation algebraically, we will follow the following four-step process:

- Isolate the absolute value expression.
- Set the quantity inside the absolute value notation equal to the positive and the negative of the quantity on the other side of the equation.
- Solve for the unknown in both equations.
- Check the answer analytically or graphically.

In order to isolate the absolute value expression, we begin by rearranging the equation to make the subject:

Recalling that the absolute value of a real number is its distance from zero, we have two possible solutions:

Either or

The solution set of the equation contains the values and 22.

We can check these solutions by substituting the values into the original equation, . In both cases, the left-hand side of the equation equals zero.

We will now consider how we can demonstrate this solution graphically. Consider the equation . The graph of this equation can be drawn on a pair of coordinate axes as follows:

As , we can draw the horizontal line onto the same graph.

Our two lines intersect at two distinct points and , which are the two solutions to the equation .

Therefore, the solution set is .

### Example 2: Solving Absolute Value Equations

Find algebraically the solution set of the equation .

### Answer

Recalling our definition of the absolute value function, we can solve this equation using piecewise notation:

As , we can consider solving the two equations and .

Firstly, when , we solve the equation

As the left-hand side of our equation is exactly the same as the right-hand side, this equation is true for all values of . However, we were restricting the solutions to values of greater than or equal to , so the solution to this part of the equation is .

Secondly, when , we need to solve the equation

In this case, as we restricted this to the interval of , this is not a valid solution to this part of the equation.

However, this solution is already covered in the solution to our first equation. The solution of the equation is , which in interval notation can be written .

We can verify this solution graphically. Let us begin by drawing the equation on the cartesian plane.

By drawing the equation on the same axes, we see that both graphs are identical for values of such that .

This confirms our earlier solution, , when .

Therefore, the solution set is .

The remaining examples in this explainer will involve solving absolute value equations that result in quadratic expressions.

### Example 3: Solving Absolute Value Equations

Find algebraically the solution set of the equation .

### Answer

In this question, we begin by noticing that since the denominator of a fraction cannot equal zero, the left-hand side of our equation is undefined when . This means that we have two different situations to consider: firstly, when and, secondly, when .

Recalling our definition of the absolute value function, we can then represent this part of the equation using piecewise notation:

When ,we solve the equation

So,

As we have already established that cannot equal , this equation has one valid solution.

When , we need to solve the equation

So,

Once again, we established that cannot equal , so this equation also has one valid solution.

The solutions of the equation are and , which can be written in set notation as .

### Example 4: Solving Absolute Value Linear Equations

Find algebraically the solution set of the equation .

### Answer

Recalling our definition of the absolute value function, we can represent the part of the equation using piecewise notation:

We can also represent the part of the equation using piecewise notation:

This means that we need to consider three scenarios.

Firstly, when , we know that and . We need to solve the equation

So, is true for all values of .

Secondly, when , we know that and . We need to solve the equation

So, is also true when .

Finally, when , we know that and . We need to solve the equation

So, there are no solutions of when .

We can therefore conclude that the equation is true for all values of less than or equal to . This can be written in set notation as .

### Example 5: Finding the Solution Set of Quadratic Equations Involving Absolute Value

Find algebraically the solution set of the equation .

### Answer

To solve this equation, we will follow the following four-step process:

- Isolate the absolute value expression.
- Set the quantity inside the absolute value notation equal to the positive and the negative of the quantity on the other side of the equation.
- Solve for the unknown in both equations.
- Check your answer analytically or graphically.

We have

Either or

Recalling that the absolute value of a real number is its distance from zero, there are two possible solutions to this equation:

Either or

Combining our two results, we have three possible solutions, , , and . This can be written as the solution set .

### Example 6: Finding the Solution Set of Root Equations Involving Absolute Value

Find the solution set of the equation .

### Answer

We begin by recalling that the absolute value of any function must be positive. This means that we can square both sides of our equation without adding any extraneous solutions:

So,

Checking that these solutions satisfy the original equation, we have

LHS: RHS: and

LHS: RHS:

We can therefore conclude that there are two solutions to the equation . They are and . This can be written as the solution set .

This can be represented graphically as shown below.

The points of intersection of and occur when and .

The solution set is, therefore, .

### Example 7: Finding the Solution Set of Quadratic Equations Involving Absolute Value by Factorization

Find algebraically the solution set of the equation .

### Answer

Recalling our definition of the absolute value function, we have two different situations to consider: firstly, when and, secondly, when .

In the first scenario, we solve the equation

So,

We check that these two solutions are valid by substituting them into the equation .

When ,

When ,

Therefore, the solutions and are both valid.

When , we solve the equation

So,

We check that these two solutions are valid by substituting them into the equation .

When ,

When ,

Therefore, the solutions and are both valid.

Combining our two results, there are four possible solutions to the equation . They are , , , and . This can be written as the solution set .

We can also demonstrate this graphically. Let us begin by sketching the equation on the Cartesian plane.

In order to sketch the graph of we reflect the portion of the graph below the -axis in the line , as shown in the figure below.

Drawing the horizontal line , we see that this intersects at four distinct points.

These correspond to the values , , , and as identified in the algebraic solution above.

Therefore, the solution set is .

We will finish this explainer by recapping some of the important concepts.

### Key Points

- The absolute value is the magnitude of a number without regard to its sign.
- We solve absolute value equations by considering the set of values for which the absolute value can be positive and negative.
- We then work out the solution sets separately and check if each solution matches the above criteria.
- Absolute value equations can be solved both graphically and algebraically.