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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 $3x−4=14?$

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b How many units away is $-5$ from $0$ on the number line?

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c Which two values are at the same distance from $2$ on the number line?

{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">6<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":2}

Challenge

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.

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

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.

- 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

The absolute value of a number $a$ is the distance between $a$ and $0$ on the number line. It is denoted as $∣a∣$ and it is always a non-negative value.
### Absolute Value Properties

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.

$∣a∣={a-a ifa≥0ifa<0 $

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 |
---|---|

Non-negativity | $∣a∣≥0$ |

Symmetry | $∣-a∣=∣a∣$ |

Idempotence | $∣∣a∣∣=∣a∣$ |

Positive-definiteness | $∣a∣=0⇔a=0$ |

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 | $∣a−b∣≤∣a−c∣+∣c−b∣$ |

Pop Quiz

Practice simplifying absolute value expressions by using the following applet.

Discussion

An absolute value equation is an equation that involves the absolute value of a variable expression.
### Number of Solutions

$∣2x−8∣=5 $

Equations of the form $∣x∣=a,$ where $a$ is a real number greater than zero, can be solved by looking for the numbers $x$ whose distance from $0$ in the number line equals $a.$ For example, the solutions of $∣x∣=4$ are all values of $x$ that are $4$ units away from $0.$
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 $ |

Method

An absolute value equation can be solved algebraically by first isolating the absolute value term. Then, consider the two possible cases for the argument inside the absolute value: one where it is positive and one where it is negative. These lead to two separate linear equations, which can be solved independently. The following example will illustrate this process.
*expand_more*
*expand_more*
*expand_more*

*expand_more*
Absolute value equations can also be solved graphically or numerically. ### Extra

Negative or Zero Constant Terms

$∣2x+4∣−16=0 $

To solve an absolute value equation, there are four steps to follow.
1

Isolate the Absolute Value Expression

Use the Properties of Equality and inverse operations to isolate the absolute value expression on one side of the equation.

2

Rewrite the Absolute Value Equation as Two Linear Equations

The absolute value function $∣x∣$ gives the distance from zero to $x,$ making it positive or zero. For an equation $∣ax+b∣=k,$ where $k≥0,$ this implies two possible cases.

$ax+b=korax+b=-k $

Applying this to the absolute value equation rewritten in Step $1$ creates two linear equations.
$2x+4=16or2x+4=-16 $

3

Solve Each Linear Equation Separately

Solve each linear equation found in Step $2$ separately.

$2x+4=16$ | $2x+4=-16$ |
---|---|

$2x=12$ | $2x=-20$ |

$x=212 $ | $x=2-20 $ |

$x=6$ | $x=-10$ |

4

Combine the Solutions

The solutions derived from the linear equations satisfy the original absolute value equation, making them solutions to the absolute value equation. Since $x=6$ and $x=-10$ are solutions for each linear equation respectively, these are also solutions to the equation $∣2x+4∣−16=0$

$Equation:Solutions: ∣2x+4∣−16=0x=6orx=-10 $

After writing the absolute value equation in the form $∣ax+b∣=c,$ where $a,$ $b,$ and $c$ are constants, check the value of $c.$

- If $c≤0,$ there is no solution. The absolute value function always returns a non-negative number, making it impossible for $∣ax+b∣$ to be negative.
- If $c=0,$ the equation simplifies to $∣ax+b∣=0,$ which has a single solution given by the linear equation $ax+b=0.$

Pop Quiz

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

Example

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 $60$ kilometers near the next wizard village, the phone will get signal. It will last until the train is $60$ kilometers past that village."
Assuming that the train does not stop in the next wizard village, for how many minutes will his smartphone stay connected? ### Hint

### Solution

### Solving the Example by Using a Number Line

Hence, the smartphone will lose connection, once again, after $70$ minutes of traveling. Taking into account that it will recover connection after the first $30$ minutes, it can be concluded that it will remain connected for $70−30=40$ minutes in total. ### Solving the Example by Using an Absolute Value Equation

It is known that Davontay's distance from the next village is $150$ kilometers. Since the train is moving, his distance is decreasing each minute by $3$ kilometers. Using this information, Davontay's distance from the city can be written in terms of the time variable $t.$
Therefore, one of the solutions to the original absolute value equation is $t=70.$ Now, the remaining equation will be solved.
Consequently, the train will be $60$ kilometers away from the city on two occasions. Once, after traveling for $30$ minutes while approaching the village. Secondly, after traveling $70$ minutes having already passed, and moving away from the village. Therefore, the phone will stay connected for a total of $70−30=40$ minutes.

Davontay looks up to check the train information monitor. He sees that the train is $150$ kilometers away from the nearest city and is moving at the speed of $3$ kilometers per minute. After how many minutes will his smartphone recover its signal?

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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.$t=sd $

▼

Substitute values and evaluate

$t=70$

$distance=150−3t $

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.
$distance=∣150−3t∣ $

Next, the times when this distance between the train and the city is exactly $60$ kilometers should be calculated. This can be done by setting up the following equation.
$∣150−3t∣=60 $

Since the constant at the right-hand side of the equation is positive, solving the equation is equivalent to solving two individual equations.
Now, the first equation will be solved.
$150−3t=-60$

$t=70$

$150−3t=60$

$t=30$

Example

Davontay is stoked to buy this video game console that is energized by an eco-friendly 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?{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mord\">$<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["25"]}}

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.

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.

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.

Closure

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.

{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">0<\/span><span class=\"mord\">\u2223<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">0<\/span><span class=\"mord\">\u2223<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">8<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">0<\/span><span class=\"mord\">\u2223<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">0<\/span><span class=\"mord\">\u2223<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

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.$ifx≥10x=12ifx<10x=8 Distance:ExampleDistance:Distance:ExampleDistance: x−10=212−10=210−x=210−8=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≥10x<10 ⇔x−10≥0⇔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) ifx−10≥0ifx−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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