10. Graphing Absolute Value Inequalities
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Chapter 2
10.
Graphing Absolute Value Inequalities
This lesson provides a deep exploration into graphing absolute value inequalities. It emphasizes the importance of understanding the boundary line, which plays a pivotal role in determining the solution set. By analyzing the relationship between two variables, one can graphically represent the solution sets of these inequalities. The lesson also touches upon real-world scenarios, such as determining the value of an object over time or interpreting the reflection of light. By mastering these concepts, one can effectively graph and interpret absolute value inequalities in various mathematical and real-world contexts.
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| | 8 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Graphing Absolute Value Inequalities
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In this lesson, linear inequalities involving two variables and absolute values will be analyzed. The solution sets of these inequalities will also be drawn and interpreted.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Graph of Absolute Value Equation in Two Variables
Consider the vertex form of absolute value equations in two variables. y= a|x- h|+ k When the vertex form of an absolute value equation is known, its graph can be drawn easily because the vertex of the graph is at ( h, k) and the graph is symmetric about x= h. Examine the graph of the indicated absolute value equation and the shaded region.
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Discussion
Absolute Value Inequalities in Two Variables
An absolute value inequality in two variables is an inequality that contains an absolute value expression and shows the relationship between two variables. If the relationship is linear, then the inequality is similar to the equation of an absolute value function. Example [-0.8em] Equation: y = |8x-3|+2 Inequality: y ≤ |8x-3|+2 While the graph of an absolute value equation is a V-shaped graph, the graph of a two-variable absolute value inequality is a region. When graphing a two-variable absolute value inequality, its boundary line plays an important role.
Method
Graphing an Absolute Value Inequality in Two Variables
The steps for graphing an absolute value inequality in two variables are similar to the steps for graphing other types of inequalities. The general method is to draw the graph of the boundary line and then determine the region to be shaded by testing a point. The following inequality will be drawn as an example. y-3 > |3x-6| To draw the graph of this inequality, the following steps can be followed.
1
Graph the Boundary Line
expand_more The boundary line of the inequality can be determined by replacing the inequality symbol with the equals sign.
Inequality: & y-3 > |3x-6| Boundary Line: & y-3 = |3x-6|
The graph of the boundary line can be drawn using a table of values. To make calculations easy, first y will be isolated.
y-3 = |3x-6| ⇕ y=|3x-6|+3
Now some points on the boundary line can be found.
| x | y=|3x-6|+3 | y |
|---|---|---|
| 0 | y=|3( 0)-6|+3 | 9 |
| 1 | y=|3( 1)-6|+3 | 6 |
| 2 | y=|3( 2)-6|+3 | 3 |
| 3 | y=|3( 3)-6|+3 | 6 |
| 4 | y=|3( 4)-6|+3 | 9 |
Plot the points and draw the boundary line. Since the given inequality is strict, the boundary line will be dashed.
2
Test a Point
expand_more Next, the region that represents the solution set for the inequality needs to be determined. To do so, choose an arbitrary point not on the boundary line and substitute it into the inequality. (0,0) is not on the boundary, so it can be used.
y -3 > |3x-6|
SubstituteII
x= 0, y= 0
0 -3 ? > | 3( 0) -6|
ZeroPropMult
Zero Property of Multiplication
0 -3 ? > | 0 -6|
SubTerms
Subtract terms
- 3 ? > |- 6|
AbsNeg
|-6|=6
- 3 > 6 *
Since - 3 > 6 is a false statement, it is not a solution to the inequality.
3
Shade the Region
expand_more The test point is not a solution, so the region that does not contain the test point — inside the V-shaped graph — represents the solution set for the inequality. Finally, the appropriate region can be shaded.
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Determining Inequality Symbol
Analyze the graph of the given absolute value inequality in two variables to determine the proper inequality symbol.
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Example
Writing an Absolute Value Inequality
Dylan writes an absolute value equation in two variables for the shape of a mirror. y-4 = - |5/2x+5| He knows that only the outside of the mirror is reflective.
a
Draw the graph of the given equation.
b
Suppose a spherical blue light source is placed above the vertex of the graph. Shade the region where the blue light can reach on the graph.
c
Write an absolute value inequality whose solution set is the shaded region, excluding the graph of the line.
Answer
a
Graph:
b
Graph:
c
y > - |5/2x+5 |+4
Hint
a
Isolate the y-variable and make a table of values.
b
What are the coordinates of the vertex of the absolute value equation? Put a point above it.
c
For strict inequalities, the points on the boundary line are not solutions. Therefore, strict inequalities have dashed boundary lines.
Solution
a
Notice that the absolute value equation for the shape of the mirror is not in vertex form.
y-4 = - |5/2x+5| The equation will be written in vertex form as follows.
y-4 = - |5/2x+5 |
▼
Rewrite
AddEqn
LHS+4=RHS+4
y = - |5/2x+5 | + 4
NumberToFrac
a = 2* a/2
y = - |5/2x+2* 5/2 | + 4
FactorOut
Factor out 5/2
y = - |5/2(x+2) | + 4
Substitute
|a* b|= |a| * |b|
y = - |5/2| * |x+2| + 4
AbsPos
|5/2|=5/2
y = - 5/2|x+2| + 4
a=- (- a)
y = - 5/2|x-(- 2)| + 4
It is now in vertex form, and its vertex is (- 2,4). Vertex Form: & y = - 5/2|x-( - 2)| + 4 [0.6em] Vertex: & ( - 2, 4) Since the x-coordinate of the vertex is - 2, a table of values will be made using some x-values below.
| x | - 5/2|x+2| + 4 | y = -5/2 |x+2 | + 4 |
|---|---|---|
| - 4 | - 5/2| - 4+2| + 4 = - 1 | - 1 |
| - 3 | - 5/2| - 3+2| + 4 = 1.5 | 1.5 |
| - 2 | - 5/2| - 2+2| + 4 = 4 | 4 |
| - 1 | - 5/2| - 1+2| + 4 = 1.5 | 1.5 |
| 0 | - 5/2| 0+2| + 4 = - 1 | - 1 |
Finally, plot the points and draw the absolute value equation.
As can be seen, the mirror has a V-shape with a vertex of (- 2,4) in the coordinate plane.
b
Now the region illuminated by a spherical blue light source is shaded when the source is placed above the vertex of the mirror. From Part A, the vertex is at (- 2,4). Therefore, the light source can be placed, for example, at (- 2,5). Then, the region illuminated by the source will be as shown.
c
The aim is to write an absolute value inequality for the region illuminated by the light source, excluding the mirror itself. Since the mirror is excluded, its graph needs to be dashed.
The shaded region represents the y-values greater than - | 52x +5 |+4. The inequality symbol will be strict because the graph of the mirror is not part of the solution set. Inequality: y > - |5/2x+5 | + 4 In this example, the mirror represents the boundary line of the inequality, the spherical light source represents a test point that satisfies the inequality, and the shaded region represents the graph of the inequality.
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Example
Graphing an Absolute Value Inequality
Ignacio buys an antique object at an auction. He plans to sell it in a few years. He describes the value of the object y (in thousands dollars) after x years as follows. y = 0.4 |x-4| + 3 He is also willing to sell the object at any time for any price greater than or equal to the value found by the above equation.
a
Write an absolute value inequality for the given situation and graph it.
b
What is the minimum price Ignacio would be willing to sell the object for 9 years from now?
Answer
a
Inequality: y ≥ 0.4|x-4|+3
Graph:
b
$5000
Hint
a
Which inequality symbol should be used? To graph the absolute value inequality, start by making a table of values for its boundary line.
b
Substitute the given value for x into the inequality found in the previous part.
Solution
a
The value of the object after x years is represented by the right-hand side of the given equation.
0.4|x-4|+3 If Ignacio can find someone who is ready to pay greater than or equal to the amount after x years, the object is sold. Therefore, the following inequality will describe the situation. y ≥ 0.4|x-4|+3 To graph this absolute value inequality, its boundary line will be drawn first. To do so, make a table of values for y=0.4|x-4|+3. Note that since x represents the number of years, it cannot be negative.
| x | y=0.4|x-4|+3 | y |
|---|---|---|
| 0 | y=0.4| 0-4|+3 | 4.6 |
| 2 | y=0.4| 2-4|+3 | 3.8 |
| 4 | y=0.4| 4-4|+3 | 3 |
| 6 | y=0.4| 6-4|+3 | 3.8 |
| 8 | y=0.4| 8-4|+3 | 4.6 |
Plot the points and draw the boundary line. Since the inequality is non-strict, the boundary line will be solid. Note that only positive values of x and y makes sense in the context of the situation.
Next, the region to be shaded will be determined. To do so, choose an arbitrary point not on the boundary line and substitute it into the inequality. Let the point be (0,0).
Since 0 ≥ 4.6 is a false statement, it is not a solution to the inequality. Therefore, the region that does not contain the test point should be shaded.
b
It is asked to find the minimum price Ignacio would be willing to sell the object for 9 years from now. To do so, 9 needs to be substituted for x into the inequality found in Part A.
This means that Ignacio will be willing to sell the object at any price greater than or equal to $5000. Therefore, $5000 is the minimum price Ignacio can accept after 9 years.
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Example
Writing an Absolute Value Inequality From a Graph
In the same auction, Emily was also interested in the object that Ignacio bought. Emily knows that Ignacio will sell it sooner or later. The maximum amount of money Emily can allocate to buy Ignacio's object is represented by the following graph.
Emily can use any amount less than $3800 for the object 4 years after the auction. Write an inequality to describe the given graph.
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Hint
To write the equation of the boundary line, use the vertex form of an absolute value equation y=a|x-h|+k, where (h,k) is the vertex of the absolute value equation.
Solution
On the given graph, x represents the years after the auction and y represents the price of the object in thousands of dollars. Since Emily can allocate any amount less than $3800 for the object 4 years after the auction, these values correspond to point (4,3.8), which is the vertex of the boundary line.
Recall the general form of an absolute value equation whose vertex is at (h,k). y = a|x-h|+k Since the vertex of the boundary line is ( 4, 3.8), the corresponding values can be substituted. y = a|x- h|+ k ⇓ y = a|x- 4|+ 3.8 To find a a point on the boundary line will be substituted. From the graph, (10,5) lies on the boundary line.
y = a|x-4|+3.8
SubstituteII
x= 10, y= 5
5 = a| 10-4|+3.8
▼
Solve for a
SubTerm
Subtract term
5 = a|6|+3.8
AbsPos
|6|=6
5 = a* 6+3.8
SubEqn
LHS-3.8=RHS-3.8
1.2= a * 6
DivEqn
.LHS /6.=.RHS /6.
0.2 = a
RearrangeEqn
Rearrange equation
a = 0.2
Now the equation of the boundary line can be written completely.
y = a|x-4|+3.8 ⇓ y = 0.2|x-4|+3.8
Since the boundary line is dashed, the inequality will be strict. The shaded region is below the boundary line. Therefore, the inequality symbol should be less than,
<.
Boundary Line: & y = 0.2|x-4|+3.8 Inequality:& y < 0.2|x-4|+3.8
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Closure
Absolute Value Inequalities on the Same Coordinate Plane
When absolute value inequalities in two variables are drawn on the same coordinate plane, they might intersect. The region where the graphs of the inequalities intersect represents the solutions that satisfy both inequalities. Consider the following absolute value inequalities. Inequality I: & y ≥ 0.4|x-4|+3 Inequality II: & y < 0.2|x-4|+3.8 There is a small region where the graphs of these inequalities intersect.
Any point in this region is the solution to both inequalities. In the context of the last two examples, Ignacio and Emily can make a deal as long as the values stay in that region.
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Graphing Absolute Value Inequalities
Exercise 2.1
Consider the absolute value inequality. y ≤ 0.5|x-3|+1 Which graph represents the solution set of the inequality?
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To graph an inequality, remember that, we always begin by determining the boundary line. The boundary line can be written by replacing the inequality sign with an equal sign. &Inequality &&Boundary Line & y ≤ 0.5|x-3|+1 && y = 0.5|x-3|+1 We have an absolute value equation for the boundary line. Notice that it is in the vertex form. Vertex Form [-1em] y= a|x- h|+ k ⇓ y= 0.5|x- 3|+ 1 The vertex is ( 3, 1). To graph the boundary line, we can make a table of values.
| x | 0.5|x-3|+1 | y = 0.5|x-3|+1 |
|---|---|---|
| 1 | 0.5| 1-3|+1 | 2 |
| 2 | 0.5| 2-3|+1 | 1.5 |
| 3 | 0.5| 3-3|+1 | 1 |
| 4 | 0.5| 4-3|+1 | 1.5 |
| 5 | 0.5| 5-3|+1 | 2 |
Let's plot the points ad draw the boundary line. Since the inequality is non-strict, the boundary line will be solid.
We will test a point to decide which region we should shade. Let the point be (0,0). If it satisfies the inequality, we will shade the region that contains the point. Otherwise, we will shade the other region.
Since the point satisfies the inequality we will shade the region that contains the point.
Therefore, the correct option is A.
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Hint & Answer
S
Solution
Which graph represents the solution set of the inequality |3x-2y| ≥ - 8 ?
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We know that all absolute value expressions are greater than or equal to zero. |3x-2y| ≥ 0 It follows then that all absolute value expressions are greater than or equal to any negative number. |3x-2y|≥ 0 ⇒ |3x-2y| ≥ - 8 This means all values of x and y will satisfy the given inequality. Therefore the solution of this inequality is all real numbers and we should shade the whole coordinate plane.
The answer is D.
Suppose that we were given an absolute value inequality, where the absolute value expression is less than a negative number. Let's change the inequality symbol of the given inequality.
|3x-2y| < - 8
Again, since all absolute value expressions are greater than or equal to zero, we cannot find any x- and y-values that satisfy the inequality |3x-2y| < - 8. Therefore, the solution set of such an inequality is the empty set.
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Solution
When graphing an inequality, the origin is often used to test which side of the boundary line to shade. For which absolute value inequalities can the origin not be used as a test point? Choose all that apply.
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We can use a test point to determine which region of the plane represents the solution set of an inequality. However, if the test point lies on the boundary line, it will give us no new information!
This is why, we prefer to use a point that is not on the boundary line. In this case, we need to determine inequalities whose boundary line passes through the origin. Let's first determine the boundary lines.
| Inequality | Boundary Line | |
|---|---|---|
| I | y > |x| | y=|x| |
| II | y ≥ |2x| | y=|2x| |
| III | y + 2 < |x| | y+2= |x| |
| IV | y+2 > |x-2| | y+2=|x-2| |
| V | y > - 2|x-2|-2 | y = - 2|x-2|-2 |
We will now substitute (0,0) into the equations of the boundary lines. If the substitution produce a true statement, it means that the boundary line passes through the origin.
| Boundary Line | Substitute (0,0) | Is it a true statement? | |
|---|---|---|---|
| I | y=|x| | 0=| 0| | Yes |
| II | y=|2x| | 0=|2* 0| | Yes |
| III | y+2= |x| | 0+2=| 0| | No |
| IV | y+2=|x-2| | 0+2=| 0-2| | Yes |
| V | y = - 2|x-2|-2 | 0=- 2| 0-2|-2 | No |
For inequalities I, II, and IV, the point (0,0) lies on the boundary line. Therefore, we will not be able to use it to determine the regions representing the solution sets of the inequalities.
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Solution
Graph the inequalities on the same coordinate plane. Inequality I: & |y| ≤ 3 Inequality II: & y ≤ - 2|x+3|+6 Which geometric shape best describes the region formed by the intersection of the solution sets of the inequalities?
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To determine the shape of the region, we should first draw each inequality separately. Then we will combine the graphs on the same coordinate plane and find their intersection. Let's start!
Inequality I
To graph |y|≤ 3, we will have to create a compound inequality first. Since |y| is less than or equal to x, the word and
will be used to form the compound inequality.
|y|≤ 3 ⇔ y≤ 3 and y≥ - 3
We can determine the boundary lines of this compound inequality by changing the inequality symbols to equal signs.
Boundary Lines
y=3 and y=- 3
Both boundary lines are horizontal. Since the inequalities are non-strict, the lines will be solid. The solution set of this inequality contains all coordinate pairs whose y-value is less than or equal to 3 and greater than or equal to - 3. This means that we should shade the region between the lines.
Also, notice that the boundary lines are parallel.
Inequality II
To determine the boundary line of the second inequality, we need to exchange the inequality symbol for an equal sign. Inequality & Boundary Line y ≤ - 2|x+3|+6 & y = - 2|x+3|+6 We see that the boundary line equation is an absolute value equation in vertex form. Vertex Form [-1em] y= a|x- h|+ k ⇓ y= - 2|x-( - 3)|+ 6 The vertex is ( - 3, 6) and its graph is symmetric about the line x= - 3. To graph the boundary line, we can make a table of values.
| x | - 2|x+3|+6 | y = - 2|x+3|+6 |
|---|---|---|
| - 6 | - 2| - 6+3|+6 | 0 |
| - 3 | - 2| - 3+3|+6 | 6 |
| 0 | - 2| 0+3|+6 | 0 |
Let's plot (- 6,0), (- 3,6), and (0,0) and draw the boundary line. Remember that the boundary line is solid since the inequality is non-strict.
Next, we decide which side of the boundary line we should shade. We can do this by testing a point that does not lie on the boundary line. If the point satisfies the inequality, it lies in the solution set. If not, we will shade the other region. Let's use ( - 3, 3).
Because (- 3,3) created a true statement, we will shade the region that contains this point.
Combining the Inequality Graphs
Let's draw the graphs of the inequalities on the same coordinate plane.
Finally, we can view only the overlapping region.
We know that the boundary lines of the first inequality are parallel. Since the overlapping region is a quadrilateral with one pair of parallel sides, it is a trapezoid.
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Solution
Graphing Absolute Value Inequalities
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