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Algebra 1 /

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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.

Variable
Expression
Number Line
Solution of an Equation
Opposite of a Number
Properties of Equality
Inverse Operations
Solving Multi-Step Equations

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

a Which of the following expressions represents the solution to the equation

3 x - 4 = 14 ?

b How many units away is

- 5

from

0

on the number line?

c Which two values are at the same distance from

2

on the number line?

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 $10$ indicates the average amount of hours a Flora battery lasts. The points at $8$ and $12$ 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

x

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 $20$ 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 $- 30$ days away!

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 $- x$ 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 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

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

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 $10$ indicates the average amount of hours a Flora battery lasts. The points at $8$ and $12$ indicate the minimum and maximum performance times, respectively.

The university wants to report the specifications of the Flora battery's performance by using an algebraic expression. Let

x

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.

As it can be seen from the diagram above, the average value of

10

is two units away from both the minimum and the maximum performance values. Note that the distance of an unknown value

x

from

10

in a number line can be calculated as a difference of those values.

if x \geq 10 x = 12 if x < 10 x = 8 Distance : Example Distance : Distance : Example Distance : x - 10 = 2 12 - 10 = 2 10 - x = 2 10 - 8 = 2

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.

x \geq 10 x < 10 \Leftrightarrow x - 10 \geq 0 \Leftrightarrow x - 10 < 0

Also, note that

10 - x

is equivalent to

- (x - 10) .

All of these observations can be summarized in the following manner.

Distance = {x - 10 - (x - 10) if x - 10 \geq 0 if x - 10 < 0

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

x

from

10

is

∣ x - 10 ∣ .

Distance = ∣ x - 10 ∣

Finally, since it is known that the

distance

to the desired values is

2,

the required equation can be set up.

2 = ∣ x - 10 ∣ ⇕ ∣ x - 10 ∣ = 2

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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Catch-Up and Review

Hint

Solution