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| 10 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
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 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.
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 |
---|---|
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∣ if b=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 |
∣∣∣x2−4∣∣∣=2 | Four | -2,2,-6,6 |
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 |
After writing the absolute value equation in the form ∣ax+b∣=c, where a, b, and c are constants, check the value of c.
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.
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 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.
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.
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.
Calculate the following absolute value expressions.
We will calculate the expression by first simplifying what is inside the absolute value. After this, we can remove the absolute value. Since the absolute value of a number is always non-negative, if the simplified expression is negative, it will become positive when the absolute value is removed. If it is already positive, it will remain unchanged.
We found that the expression simplifies to 5.
As in Part A, we start by simplifying the expression inside the absolute value. Then we remove the absolute value and change the sign of the expression to to positive if the simplified expression inside the absolute value is negative.
We found that the expression simplifies to 2.
Let's follow the same procedure to simplify the expression and then remove the absolute value. Remember, the absolute value of an expression is never negative.
We found that the expression simplifies to 1.
Consider the following number lines.
When we know two points on a number line and their midpoint, we can write an absolute value equation. The equation is written on the following format. |x- midpoint|= distance to midpoint Let's start by rewriting all equations to match this format. i. & |x-( -2) |= 4 [0.5em] ii. & |x- 4|= 2 [0.5em] iii. & |x-( -4)|= 2 [0.5em] iv. & |x- 2|= 4 Next, we will identify the midpoint and distance to each point in each of the graphs. After that, we can pair the graph with the correct equation. Let's begin with graph A.
With the information of the graph, we can write the absolute value equation of graph A. |x-( -4)| = 2 Therefore, graph A is associated with equation iii. Now let's take a look at graph B.
Now we can use this information to write the equation of graph B. |x-( -2)| = 4 Therefore, graph B is associated with equation i. Now we can go to graph C.
Using the information obtained, we can also write the equation of this graph. |x- 2| = 4 Therefore, graph C is associated with equation iv. Finally, we can look at graph D.
Now we can write the equation. |x- 4| = 2 Graph D is associated with equation ii. Now let's take a final look of all the pairings! A& → -1pt &&iii. B& → -1pt &&i. C& → -1pt &&iv. D& → -1pt &&ii.
Simplify the following absolute value expressions.
Let's start by calculating all of the individual absolute values. Then we can add the results. Remember, since an absolute value is always non-negative, any argument that is negative changes signs and becomes positive.
Now that we found all the absolute values, we can add all the terms! 1+4+11 = 16
Again, we start by calculating all the individual absolute values. Remember that if the argument of an absolute value is negative, we need to change the sign for the argument to become positive.
Now we can add or subtract the results regularly. 99 - 17 + 180 = 262
As in previous parts, we will evaluate the absolute values and simplify the results. But we have to be careful, since if the result of an absolute value has a negative sing, the result is a negative number.
Now we can add and subtract the numbers regularly. - 2 + 11.5 - 7.5 = 2 Now we have finished. Good job!
Solve the absolute value equations. Write the answers from least to greatest if needed.
An absolute value measures an expression's distance from a midpoint on a number line. |m+4|= 7 This equation means that the distance is 7, either in the positive direction or in the negative direction. Algebraically, we can separate the absolute value equation into two cases after the absolute value is removed, one case where 7 is positive and the other where it is negative. |m+4|= 7 ⇒ lm+4= 7 m+4= - 7 To find the solutions to the absolute value equation, we need to solve both of these cases for m.
We can see that m has to possible values. Let's write these answers from least to greatest. m_1 & = -11 m_2 & =3
As in Part A, we get two cases when we remove the absolute value.
Now we can see that x has to possible values. Let's write these answers from least to greatest. x_1 & = -7 x_2 & =23
As in previous parts, we get two cases when we remove the absolute value.
Again we can see that x has to possible values. Let's write these answers from least to greatest. x_1 & = -5 x_2 & =5
As in previous parts, we get two cases when we remove the absolute value.
This time we can also see that b has to possible values. Let's write these answers from least to greatest. b_1 & = -7 b_2 & =12
Before we can solve this equation, we need to isolate the absolute value expression.
An absolute value measures the distance from a midpoint on a number line. Note that a distance cannot be negative. This means the absolute value of a number must also be non-negative. |y-1|≠ -3 This absolute value equation has no solution.
Before an election, a poll shows that 55% of voters are likely to vote for Candidate A. This has a margin of error of ±3 percentage points.
Solve the equation. Write the solutions from least to greatest.
Let's label the percentage of voters that are likely to vote for Candidate A v. The poll tells us that 55 % of voters, including a margin of error of 3 %, are likely to vote for the candidate. A margin of error indicates that the difference between the actual value v and 55 is ± 3. v - 55 = ± 3 The ± symbol indicates that the subtraction can be equal to -3 or 3. Note that we can rewrite this expression as two equations. v-55=-3 v-55= 3 If we look closely at these equations, we can see that this is how we separate the cases of an absolute value equation. Let's write the absolute value equation that results in these cases! |v-55|=3
In order to solve the absolute value equation, we start by to separating the equation back into its cases from Part A. Then we can solve each case equation individually. Let's do it!
Looking at the results, we can see that the least percent of voters who will vote for Candidate A is 52 %, and the greatest percent is 58 %.