assignment 1 number order and absolute value

3.6 Absolute Value Functions

Learning objectives.

In this section, you will:

• Graph an absolute value function.
• Solve an absolute value equation.

Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will continue our investigation of absolute value functions .

Understanding Absolute Value

Recall that in its basic form f ( x ) = | x | , f ( x ) = | x | , the absolute value function is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign. Knowing this, we can use absolute value functions to solve some kinds of real-world problems.

Absolute Value Function

The absolute value function can be defined as a piecewise function

Using Absolute Value to Determine Resistance

Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often ±1%, ± 5%, ±1%, ± 5%, or ± 10% . ± 10% .

Suppose we have a resistor rated at 680 ohms, ± 5 % . ± 5 % . Use the absolute value function to express the range of possible values of the actual resistance.

We can find that 5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance R R in ohms,

Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.

Graphing an Absolute Value Function

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in Figure 2 .

Figure 3 shows the graph of y = 2 | x – 3 | + 4. y = 2 | x – 3 | + 4. The graph of y = | x | y = | x | has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at ( 3 , 4 ) ( 3 , 4 ) for this transformed function.

Writing an Equation for an Absolute Value Function Given a Graph

Write an equation for the function graphed in Figure 4 .

The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. See Figure 5 .

We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance as shown in Figure 6 .

From this information we can write the equation

Note that these equations are algebraically equivalent—the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression. Note also that if the vertical stretch factor is negative, there is also a reflection about the x-axis.

If we couldn’t observe the stretch of the function from the graphs, could we algebraically determine it?

Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for x x and f ( x ) . f ( x ) .

Now substituting in the point (1, 2)

Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically reflected, and vertically shifted up 3 units.

Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?

Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.

No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see Figure 7 ).

Solving an Absolute Value Equation

In Other Type of Equations , we touched on the concepts of absolute value equations. Now that we understand a little more about their graphs, we can take another look at these types of equations. Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as 8 = | 2 x − 6 | , 8 = | 2 x − 6 | , we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.

Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,

Solutions to Absolute Value Equations

For real numbers A A and B B , an equation of the form | A | = B , | A | = B , with B ≥ 0 , B ≥ 0 , will have solutions when A = B A = B or A = − B . A = − B . If B < 0 , B < 0 , the equation | A | = B | A | = B has no solution.

Given the formula for an absolute value function, find the horizontal intercepts of its graph .

• Isolate the absolute value term.
• Use | A | = B | A | = B to write A = B A = B or −A = B , −A = B , assuming B > 0. B > 0.
• Solve for x . x .

Finding the Zeros of an Absolute Value Function

For the function f ( x ) = | 4 x + 1 | − 7 , f ( x ) = | 4 x + 1 | − 7 , find the values of x x such that f ( x ) = 0. f ( x ) = 0.

The function outputs 0 when x = 3 2 x = 3 2 or x = − 2. x = − 2. See Figure 8 .

For the function f ( x ) = | 2 x − 1 | − 3 , f ( x ) = | 2 x − 1 | − 3 , find the values of x x such that f ( x ) = 0. f ( x ) = 0.

Should we always expect two answers when solving | A | = B ? | A | = B ?

No. We may find one, two, or even no answers. For example, there is no solution to 2 + | 3 x − 5 | = 1. 2 + | 3 x − 5 | = 1.

Access these online resources for additional instruction and practice with absolute value.

• Graphing Absolute Value Functions
• Graphing Absolute Value Functions 2

3.6 Section Exercises

How do you solve an absolute value equation?

How can you tell whether an absolute value function has two x -intercepts without graphing the function?

When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

How can you use the graph of an absolute value function to determine the x -values for which the function values are negative?

Describe all numbers x x that are at a distance of 4 from the number 8. Express this set of numbers using absolute value notation.

Describe all numbers x x that are at a distance of 1 2 1 2 from the number −4. Express this set of numbers using absolute value notation.

Describe the situation in which the distance that point x x is from 10 is at least 15 units. Express this set of numbers using absolute value notation.

Find all function values f ( x ) f ( x ) such that the distance from f ( x ) f ( x ) to the value 8 is less than 0.03 units. Express this set of numbers using absolute value notation.

For the following exercises, find the x - and y -intercepts of the graphs of each function.

f ( x ) = 4 | x − 3 | + 4 f ( x ) = 4 | x − 3 | + 4

f ( x ) = − 3 | x − 2 | − 1 f ( x ) = − 3 | x − 2 | − 1

f ( x ) = − 2 | x + 1 | + 6 f ( x ) = − 2 | x + 1 | + 6

f ( x ) = − 5 | x + 2 | + 15 f ( x ) = − 5 | x + 2 | + 15

f ( x ) = 2 | x − 1 | − 6 f ( x ) = 2 | x − 1 | − 6

f ( x ) = | − 2 x + 1 | − 13 f ( x ) = | − 2 x + 1 | − 13

f ( x ) = − | x − 9 | + 16 f ( x ) = − | x − 9 | + 16

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

y = | x − 1 | y = | x − 1 |

y = | x + 1 | y = | x + 1 |

y = | x | + 1 y = | x | + 1

For the following exercises, graph the given functions by hand.

y = | x | − 2 y = | x | − 2

y = − | x | y = − | x |

y = − | x | − 2 y = − | x | − 2

y = − | x − 3 | − 2 y = − | x − 3 | − 2

f ( x ) = − | x − 1 | − 2 f ( x ) = − | x − 1 | − 2

f ( x ) = − | x + 3 | + 4 f ( x ) = − | x + 3 | + 4

f ( x ) = 2 | x + 3 | + 1 f ( x ) = 2 | x + 3 | + 1

f ( x ) = 3 | x − 2 | + 3 f ( x ) = 3 | x − 2 | + 3

f ( x ) = | 2 x − 4 | − 3 f ( x ) = | 2 x − 4 | − 3

f ( x ) = | 3 x + 9 | + 2 f ( x ) = | 3 x + 9 | + 2

f ( x ) = − | x − 1 | − 3 f ( x ) = − | x − 1 | − 3

f ( x ) = − | x + 4 | − 3 f ( x ) = − | x + 4 | − 3

f ( x ) = 1 2 | x + 4 | − 3 f ( x ) = 1 2 | x + 4 | − 3

Use a graphing utility to graph f ( x ) = 10 | x − 2 | f ( x ) = 10 | x − 2 | on the viewing window [ 0 , 4 ] . [ 0 , 4 ] . Identify the corresponding range. Show the graph.

Use a graphing utility to graph f ( x ) = − 100 | x | + 100 f ( x ) = − 100 | x | + 100 on the viewing window [ − 5 , 5 ] . [ − 5 , 5 ] . Identify the corresponding range. Show the graph.

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

f ( x ) = − 0.1 | 0.1 ( 0.2 − x ) | + 0.3 f ( x ) = − 0.1 | 0.1 ( 0.2 − x ) | + 0.3

f ( x ) = 4 × 10 9 | x − ( 5 × 10 9 ) | + 2 × 10 9 f ( x ) = 4 × 10 9 | x − ( 5 × 10 9 ) | + 2 × 10 9

For the following exercises, solve the inequality.

If possible, find all values of a a such that there are no x - x - intercepts for f ( x ) = 2 | x + 1 | + a . f ( x ) = 2 | x + 1 | + a .

If possible, find all values of a a such that there are no y y -intercepts for f ( x ) = 2 | x + 1 | + a . f ( x ) = 2 | x + 1 | + a .

Real-World Applications

Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and x x represents the distance from city B to city A, express this using absolute value notation.

The true proportion p p of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable x x for the score.

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using x x as the diameter of the bearing, write this statement using absolute value notation.

The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is x x inches, express the tolerance using absolute value notation.

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• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: College Algebra 2e
• Publication date: Dec 21, 2021
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/college-algebra-2e/pages/3-6-absolute-value-functions

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Absolute Value and Ordering Numbers by Value

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Understand ordering and absolute value of rational numbers.

More specific topics in understand ordering and absolute value of rational numbers..

• Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.
• Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
• Distinguish comparisons of absolute value from statements about order.

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• Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
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• Terms of Use

Absolute Value

Our Absolute Value lesson plan teaches students about absolute value, including what it is and what the symbols used to designate absolute zero are. Students also learn how the absolute value symbols are interpreted in equations and that absolute values are always positive because they measure distance.

Included with this lesson are some adjustments or additions that you can make if you’d like, found in the “Options for Lesson” section of the Classroom Procedure page. One of the optional additions to this lesson is to hand out index cards with problems and then have students place themselves on a number line for their answer or order themselves according to their answer from greatest to least.

Description

Additional information, what our absolute value lesson plan includes.

Lesson Objectives and Overview: Absolute Value defines absolute value, teaches the symbols used to designate absolute zero and how they are interpreted in equations, and explains that absolute values are always positive because they measure distance. At the end of the lesson, students will be able to understand absolute value and how to solve expressions using absolute value bars as grouping symbols. This lesson is for students in 5th grade and 6th grade.

Classroom Procedure

Every lesson plan provides you with a classroom procedure page that outlines a step-by-step guide to follow. You do not have to follow the guide exactly. The guide helps you organize the lesson and details when to hand out worksheets. It also lists information in the blue box that you might find useful. You will find the lesson objectives, state standards, and number of class sessions the lesson should take to complete in this area. In addition, it describes the supplies you will need as well as what and how you need to prepare beforehand. The only supplies you will need are decks of cards.

Options for Lesson

Included with this lesson is an “Options for Lesson” section that lists a number of suggestions for activities to add to the lesson or substitutions for the ones already in the lesson. To add to the lesson activity, you could h ave students increase the number of rows for the game or add in your own deck of cards with larger numbers. For an additional activity, you could  hand out index cards with problems and then have students place themselves on a number line for their answer or order themselves according to their answer from greatest to least. Finally, you can provide challenge questions using fractions or decimals.

Teacher Notes

The teacher notes page includes lines that you can use to add your own notes as you’re preparing for this lesson.

ABSOLUTE VALUE LESSON PLAN CONTENT PAGES

The Absolute Value lesson plan includes two pages of content. All numbers except zero have an absolute value. The symbol for absolute value is two straight lines around the number or expression. For example, |9| = 9 means the absolute value of 9 is 9, and |-8| = 8 means the absolute value of -8 is 8.

It’s important to pay attention to whether the negative is placed inside or outside of the absolute value bars. When it’s inside, the absolute value is positive. When it’s outside, the absolute value is negative.

Absolute value is the distance from zero on a number line. Absolute value is a measure of distance and is therefore always positive. The lesson includes two number lines to illustrate this. The absolute value of 0 is zero, because zero is not positive or negative.

We can also think of the absolute value bars as a grouping symbol, like parenthesis. When we use the order of operations to solve an expression or equation, we solve anything inside absolute value bars before doing any other operation. The lesson includes several examples that illustrate this. The first is: |-6 + 2| = |-4| = 4 .

If there is more than one set of absolute value bars, you complete the operations inside the sets of bars before completing any other operations. In the order of operations, the first set is grouping symbols. The lesson includes more examples to illustrate this. In one example, we’re solving the following problem using these steps: |-3 + 1| + |5 + 2| = |-4| + |7| = 4 + 7 = 11 .

The lesson closes with a few more examples to help students practice using these concepts.

ABSOLUTE VALUE LESSON PLAN WORKSHEETS

The Absolute Value lesson plan includes three worksheets: an activity worksheet, a practice worksheet, and a homework assignment. You can refer to the guide on the classroom procedure page to determine when to hand out each worksheet.

PLAYING ZERO ACTIVITY WORKSHEET

Students will complete the lesson activity in groups. To play the game, each player gets two cards, one face up and one down. All the students at the table can see the face up cards. Black cards are positive while red cards are negative. Each player can get additional cards by asking them to “hit me”. They can have up to four cards each and must use all of them. For each round, the person with the total closest to zero wins that round.

At the end of the game, students will take the absolute value of their “total” column for each hand and will add them together. The person closest to zero in the absolute value column wins the game.

ABSOLUTE VALUE PRACTICE WORKSHEET

The practice worksheet asks students to complete two short exercises. For the first, they will find the absolute value of different numbers. For the second, they will simplify expressions, showing their work.

MATCHING HOMEWORK ASSIGNMENT

For the homework assignment, students will match nine problems to their correct answers. They will then either find the absolute value or simplify each equation.

This lesson also includes a quiz that you can use to test students’ understanding of the lesson material. For the quiz, students will find the absolute value of six numbers and will then simplify six expressions, showing their work.

Worksheet Answer Keys

This lesson plan includes answer keys for the practice worksheet, the homework assignment, and the quiz.  If you choose to administer the lesson pages to your students via PDF, you will need to save a new file that omits these pages. Otherwise, you can simply print out the applicable pages and keep these as reference for yourself when grading assignments.

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Absolute Value of a Number

Think of the distance from one integer to another as being one step. If you start at zero and take two steps to the right, you get to 2. If you start at 0 and take two steps to the left, you get to -2. If you take two steps forward, we take two steps back. We come together, ‘cuz opposites attract. In any case, we go the same distance.

Each integer (except 0) has two pieces of information: its distance from 0, and its direction from 0.

The absolute value of a number is the distance of a number from zero on the number line. The absolute value is always positive since a distance is non-negative. The absolute value of an integer is the numerical value without regard to whether the sign is negative or positive. On a number line, it is the distance between the number and zero.

The absolute value of -15 is 15. The absolute value of +15 is 15

The symbol for absolute value is to enclose the number of vertical bars such as |-20| = 20 and read “The absolute value of -20 equals 20”.

Examples on absolute value of an integer:

(i) Absolute value of – 7 is written as |- 7| = 7 [here mod of – 7 = 7]

(ii) Absolute value of + 2 is written as |+ 2| = 2 [here mod of + 2 = 2]

(iii) Absolute value of – 15 is written as |- 15| = 15 [here mod of – 15 = 15]

(iv) Absolute value of + 17 is written as |+ 17| = 17 [here mod of + 17 = 17]

Find the mod of:

(i) |14 – 6| = |8| = 8

(ii) – |- 10| = – 10

(iii) 15 – |- 6| = 15 – 6 = 9

(iv) 7 + |- 7| = 7 + 7 = 14

Note: Taking the negative of a number doesn’t always give us a negative number, as the previous examples demonstrate. So before assuming a number is negative just because you see a negative sign, make sure that there’s only one of them. If there are multiple negative signs, the number may be negative or positive, depending on how many negative signs there are. We know that you may not don’t never can’t want to deal with too many negative signs, but it’s just a part of life.

Information Source:

• www.aaamath.com
• www.math-only-math.com

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1.9: Absolute Value Functions

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• Page ID 59844

• David Arnold
• College of the Redwoods

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• 1.9.1: Piecewise-Defined Functions In preparation for the definition of the absolute value function, it is extremely important to have a good grasp of the concept of a piecewise-defined function.
• 1.9.2: Absolute Value The absolute value of a number is a measure of its magnitude, sans (without) its sign.
• 1.9.3: Absolute Value Equations
• 1.9.4: Absolute Value Inequalities

Thumbnail: The graph of the absolute value function for real numbers. (CC BY-SA 3.0; Qef and Ævar Arnfjörð Bjarmason via Wikipedia).