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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 university is developing an ecofriendly 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.
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.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.
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.
There are several properties and identities that are useful when simplifying expressions or solving equations dealing with absolute values. For any two real numbers $a$ and $b,$ the following relationships and identities hold true.
Property  Algebraic Representation 

Nonnegativity  $∣a∣≥0$ 
Symmetry  $∣a∣=∣a∣$ 
Idempotence  $∣∣a∣∣=∣a∣$ 
Positivedefiniteness  $∣a∣=0⇔a=0$ 
Identity of Indiscernibles  $∣a−b∣=0⇔a=b$ 
Multiplicativity  $∣ab∣=∣a∣⋅∣b∣$ 
Preservation of Division  $∣∣∣∣ ba ∣∣∣∣ =∣b∣∣a∣ ifb =0$ 
Subadditivity  $∣a+b∣≤∣a∣+∣b∣$ 
Triangle Inequality  $∣a−b∣≤∣a−c∣+∣c−b∣$ 
Practice simplifying absolute value expressions by using the following applet.
Since there are two points on the number line that fulfill this requirement, there are two solutions to the equation $∣x∣=4,$ namely $x=4$ and $x=4.$ However, solving an absolute value equation, in general, might require a more elaborate and structured approach.
Simple absolute value equations of the form $∣x∣=a,$ can have no, one, or two solutions, depending on the value of $a.$ However, more complex absolute value equations may have more than two solutions.
Equation  Number of Solutions  Solution(s) 

$∣x∣=4$  Zero  No solution 
$∣x∣=0$  One  $0$ 
$∣x∣=4$  Two  $4,4$ 
$∣∣∣ x_{2}−4∣∣∣ =2$  Four  $2 ,2 ,6 ,6 $ 
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.
$2x+4>0$  $2x+4<0$ 

$∣2x+4∣=2x+4$  $∣2x+4∣=(2x+4)$ 
$∣2x+4∣=16⇓2x+4=16 $

$∣2x+4∣=16⇓(2x+4)=16⇕2x+4=16 $

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.
Practice solving absolute value equations by using the following applet. Indicate which number line represents the solution set of the given equation.
The current distance from the train to the village is $150$ kilometers, and for each passing minute its distance to the next village is reduced by $3$ kilometers. When will the trains distance be equal to $60$ kilometers?
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.
It is helpful to summarize the important information in a diagram. Recall that the train is originally $150$ 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 $60$ kilometers.
From the diagram, it can be seen that the train needs to cover a distance of $150−60=90$ kilometers for the smartphone to recover its signal. By using the speed formula, the time of travel for that distance can be calculated. Therefore, the phone will recover its signal after $30$ minutes. Now, to determine how long it will keep connected, the time the train will take to leave the $60$ kilometers proximity from the wizard village should be found. As can be seen, the train needs to cover a distance of $150+60=210$ kilometers for the smartphone to lose connection again. Now the formula for speed will be used once more. Recall that $t$ was already isolated in the previous calculations. This result will be reused.Davontay is stoked to buy this video game console that is energized by an ecofriendly battery — everyone wants one. Davontay has $$280$ saved for this console, but its average selling price is $$350.$ 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 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?The value on the righthand side of the equation is positive. Therefore, solving the absolute value equation is equivalent to solving two individual equations.
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 lefthand side of the equation, and the quantity on the righthand side is positive. Therefore, solving this absolute value equation is equivalent to solving two individual onestep equations.
As has been determined, according to the model, the minimum price at which the console can be purchased online is $$305.$ The difference between the minimum price and what Dylan has saved $$285$ is $$25.$ Therefore, Davontay needs to save $$25$ more to be able to buy the game console he has long awaited.
A university is developing an ecofriendly 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.
What is the distance from the average performance value to the minimum and maximum values? Use this distance to form an absolute value equation.
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.