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Image Credits expand_more
Mathematical models, at times, can predict irrelevant values when describing real-world scenarios. For example, the calculation of a length only makes sense to obtain a non-negative result. Thankfully, this lesson will show how situations that require a positive result can be modeled using the absolute value. Additionally, equations that involve absolute value expressions will be explored.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Here are a few practice exercises before getting started with this lesson.

a Which of the following expressions represents the solution to the equation
b How many units away is from on the number line?
Number line showing two points at 0 and -5, respectively
c Which two values are at the same distance from on the number line?
Number line showing a point at 2

Challenge

Modeling the Performance of a Battery

A university is developing an eco-friendly battery for tablets called Flora, which uses no harmful chemicals for the environment. After running some tests, the following number line describes the results. Fully charged, the point at indicates the average amount of hours a Flora battery lasts. The points at and indicate the minimum and maximum performance times, respectively.

Number line showing the average, minimum, and maximum amount of hours of the battery performance
The university wants to report Flora's performance by using an algebraic expression. Let represent the number of hours the battery can be used when fully charged. Then, find an equation that models the situation and whose solutions are the minimum and the maximum hours the battery can last.

Explore

Why and How to Define an Always Positive Quantity

Think of a mathematical model that needs to predict a strictly positive quantity. Ever wonder exactly how many days until the end of school, but are only given the calendar date? Well, consider a formula that counts the number of days. A date is entered into the formula and the prediction of days away from today is made.

Number line showing two calendars, the predicted date and how far it is from today

Then, another date is entered, but this time the formula says that it is days away!

Number line showing two calendars, the predicted date and how far it is from today

Considering the given information about the formula, try to answer the following questions.

  • Since days cannot be negative, what can the minus sign mean here?
  • How many days away is this date from today?
  • If the formula predicted a value of for a specific date, how many days away would it be from today?
  • Is there a reasonable way to assign a positive value to every real number in general?

Discussion

Absolute Value

The absolute value of a number is the distance between and on the number line. It is denoted as and it is always a non-negative value.
Interactive number line illustrating the concept of absolute value
The absolute value is defined for any real number. The absolute value of a negative number is its opposite value, while the absolute value of a positive number is equal to itself.

Absolute Value Properties

There are several properties and identities that are useful when simplifying expressions or solving equations dealing with absolute values. For any two real numbers and the following relationships and identities hold true.

Property Algebraic Representation
Non-negativity
Symmetry
Idempotence
Positive-definiteness
Identity of Indiscernibles
Multiplicativity
Preservation of Division
Subadditivity
Triangle Inequality
The absolute value is useful in many situations when the quantity to be modeled or described mathematically is known to be positive. For example, the absolute value is often used when calculating distances, lengths, age, time, area, etc.

Pop Quiz

Simplifying Absolute Value Expressions

Practice simplifying absolute value expressions by using the following applet.

Interactive applet showing different absolute value expressions

Discussion

Absolute Value Equations

An absolute value equation is an equation that involves the absolute value of a variable expression.
Equations of the form where is a real number greater than zero, can be solved by looking for the numbers whose distance from in the number line equals For example, the solutions of are all values of that are units away from
Number line showing the solutions to the absolute value equation |x|=4

Since there are two points on the number line that fulfill this requirement, there are two solutions to the equation namely and However, solving an absolute value equation, in general, might require a more elaborate and structured approach.

Number of Solutions

Simple absolute value equations of the form can have no, one, or two solutions, depending on the value of However, more complex absolute value equations may have more than two solutions.

Equation Number of Solutions Solution(s)
Zero No solution
One
Two
Four

Method

Solving an Absolute Value Equation

An absolute value equation can be solved algebraically by applying the definition of absolute value and considering the possible values for the argument of the absolute value. Consider the following example equation.
To solve an absolute value equation, there are five steps to follow.
1
Isolate the Absolute Value Expression
expand_more
By using the properties of equality and inverse operations, isolate the absolute value expression.
AddEqn

2
Analyze the Constant Term on the Right-Hand Side
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There are three possibilities for the constant on the right-hand side of the equation; it can be positive, negative, or zero. In the example being studied, this term is
Nevertheless, in the next step, every case will be considered to illustrate how to proceed in every circumstance. The value of this term has a direct impact on the number of solutions of the absolute value equation.
3
Consider All Possibilities for The Quantity Inside the Absolute Value
expand_more

Positive Constant Term

If the constant term is positive, there are two possibilities for the argument of the absolute value: it can be either positive or negative. If it is positive, by the definition of an absolute value, no change is needed. However, if it is negative, the definition of an absolute value requires using the opposite of the value instead.

Therefore, solving the original absolute value equation is equivalent to solving two individual equations that no longer involve an absolute value expression. These individual equations can be solved using any preferred method.

Solving the original absolute value equation |2x+4|=16 is equivalent to solving the individual equivalent equations 2x+4=-16 and 2x+4=16.

Zero Constant Term

In the particular case were the constant term is zero, there is only one possibility for the quantity inside the absolute value. For the given example equation, imagine for a moment that the right-hand side of the equation is instead of
By the properties of the absolute value the quantity inside must be equal to zero. Therefore, there is only one equation to solve.

Negative Constant Term

Another special case is when the quantity on the right-hand side is negative. For example, consider now that the right-hand side of the equation were instead of
By the definition of absolute value, any absolute value expression is non-negative. Therefore, in cases like this, there is no solution to the absolute value equation.
4
Solve the Equivalent Equations
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At this point, all that is left is to solve the obtained equations which no longer involve an absolute value expression. This can be done using any convenient method. Solve the first equation.
Solve for
SubEqn

DivEqn

Now, solve the remaining equation.
Solve for
SubEqn

DivEqn

5
Combine All of the Solutions
expand_more
The solution set for the original absolute value equation is equals the union of the solution sets of the equivalent equations
Absolute value equations can also be solved graphically or numerically.

Pop Quiz

Practice Solving Absolute Value Equations

Practice solving absolute value equations by using the following applet. Indicate which number line represents the solution set of the given equation.

Interactive applet showing different absolute value equations and different number lines representing possible solutions

Example

Using Absolute Value Equations to Model Real Life Situations

Davontay is on a train trip to a school of magic. His smartphone is powered by an eco-friendly battery, but still a young product, it easily loses reception. A vendor notices and gives him hope. She exclaims, "Don't worry, lad. When the train is kilometers near the next wizard village, the phone will get signal. It will last until the train is kilometers past that village."
Animation showing the train motion and the smartphone gaining a losing connection
Davontay looks up to check the train information monitor. He sees that the train is kilometers away from the nearest city and is moving at the speed of kilometers per minute. After how many minutes will his smartphone recover its signal?
Assuming that the train does not stop in the next wizard village, for how many minutes will his smartphone stay connected?

Hint

The current distance from the train to the village is kilometers, and for each passing minute its distance to the next village is reduced by kilometers. When will the trains distance be equal to kilometers?

Solution

This scenario will be solved using two methods. First, by using a number line, and then, it will be solved through setting up and solving an absolute value equation.

Solving the Example by Using a Number Line

It is helpful to summarize the important information in a diagram. Recall that the train is originally kilometers away from the next wizard village, and the smartphone will only get a signal when the distance of the train from the village is equal to kilometers.

Diagram showing the distance from the train and the city with a number line for reference, and indicating the distance to be traveled for the smartphone to recover connection
From the diagram, it can be seen that the train needs to cover a distance of kilometers for the smartphone to recover its signal. By using the speed formula, the time of travel for that distance can be calculated.
Solve for
MultEqn

DivEqn

SubstituteII

,

CalcQuot

Calculate quotient

Therefore, the phone will recover its signal after minutes. Now, to determine how long it will keep connected, the time the train will take to leave the kilometers proximity from the wizard village should be found.
Diagram showing the distance from the train and the city with a number line for reference, and indicating the distance to be traveled for the smartphone to lose connection
As can be seen, the train needs to cover a distance of kilometers for the smartphone to lose connection again. Now the formula for speed will be used once more. Recall that was already isolated in the previous calculations. This result will be reused.
Substitute values and evaluate
SubstituteII

,

CalcQuot

Calculate quotient

Hence, the smartphone will lose connection, once again, after minutes of traveling. Taking into account that it will recover connection after the first minutes, it can be concluded that it will remain connected for minutes in total.

Solving the Example by Using an Absolute Value Equation

It is known that Davontay's distance from the next village is kilometers. Since the train is moving, his distance is decreasing each minute by kilometers. Using this information, Davontay's distance from the city can be written in terms of the time variable
However, since the distance from the village is a length, it must be a non-negative quantity. This can be be assured by taking the absolute value of the expression.
Next, the times when this distance between the train and the city is exactly kilometers should be calculated. This can be done by setting up the following equation.
Since the constant at the right-hand side of the equation is positive, solving the equation is equivalent to solving two individual equations.
Solving the original absolute value equation |150-3t| = 60 is equivalent to solving the individual equations 150-3t = -60 and 150-3t = 60.
Now, the first equation will be solved.
Solve for
SubEqn

SubTerm

Subtract term

DivEqn

Therefore, one of the solutions to the original absolute value equation is Now, the remaining equation will be solved.
Solve for
SubEqn

SubTerm

Subtract term

DivEqn

Consequently, the train will be kilometers away from the city on two occasions. Once, after traveling for minutes while approaching the village. Secondly, after traveling minutes having already passed, and moving away from the village. Therefore, the phone will stay connected for a total of minutes.

Example

Using Absolute Value Equations to Define a Range of Prices

Davontay is stoked to buy this video game console that is energized by an eco-friendly battery — everyone wants one. Davontay has saved for this console, but its average selling price is Shopping online, he uses a search program designed to find discounts. He finds that the differences in prices can be modeled with an absolute value equation.

The fluctuations in the console price can be modeled with the absolute value equation |x-350| = 45
The solutions for this absolute value equation represent the minimum and maximum prices for the console found online. Davontay is planning to buy it at the lowest price. How much more money does Davontay need to save so he can afford the lowest price found online according to this model?

Hint

The value on the right-hand side of the equation is positive. Therefore, solving the absolute value equation is equivalent to solving two individual equations.

Solution

First, the absolute value equation will be solved to find the minimum selling price. Note that the absolute value expression is already isolated on the left-hand side of the equation, and the quantity on the right-hand side is positive. Therefore, solving this absolute value equation is equivalent to solving two individual one-step equations.

Solving the original absolute value equation |x-350| = 45 is equivalent to solving the individual one-step equations x-350 = -45 and x-350 = 45.

As has been determined, according to the model, the minimum price at which the console can be purchased online is The difference between the minimum price and what Dylan has saved is Therefore, Davontay needs to save more to be able to buy the game console he has long awaited.

It has been seen how an absolute value equation can have multiple solutions. Finally, the challenge at the beginning of the lesson can be solved using an absolute value equation.

Closure

Using an Absolute Value Equation to Model Battery Performance

A university is developing an eco-friendly battery for tablets called Flora that uses no harmful chemicals for the environment. After running some tests, the following number line describes the results. Fully charged, the point at indicates the average amount of hours a Flora battery lasts. The points at and indicate the minimum and maximum performance times, respectively.

Number line showing the average, minimum, and maximum amount of hours of the battery performance
The university wants to report the specifications of the Flora battery's performance by using an algebraic expression. Let represent the number of hours the battery can be used when fully charged. Then, find an equation that models the situation and whose solutions are the minimum and the maximum hours the battery can last.

Hint

What is the distance from the average performance value to the minimum and maximum values? Use this distance to form an absolute value equation.

Solution

To set up an absolute value equation having the required maximum and minimum values as the solutions, it is useful to identify what is the distance from them to the average value on the number line.

Number line indicating the distances of the minimum and maximum performance values from the average one
As it can be seen from the diagram above, the average value of is two units away from both the minimum and the maximum performance values. Note that the distance of an unknown value from in a number line can be calculated as a difference of those values.
Next, to write the distance in the form of an absolute value expression, the inequalities will be rewritten to have zeros on the right-hand sides resemble the definition for the absolute value of a quantity.
Also, note that is equivalent to All of these observations can be summarized in the following manner.
Notice that this formula for the distance is now very similar to the definition of the absolute value of a number. Therefore, all the previous information imply that the distance for a value from is
Finally, since it is known that the to the desired values is the required equation can be set up.
Now that the Flora battery has been tested and the performance results can be reported using a formal mathematical expression, the university can let everyone know about Flora to power devices in a cleaner way!
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