1 Lesson 2.2.2 Using Absolute Value. 2 Lesson 2.2.2 Using Absolute Value California Standards: Number Sense 2.5 Understand the meaning of the absolute

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Lesson 2.2.2Lesson 2.2.2

Using Absolute ValueUsing Absolute Value

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Lesson

2.2.2Using Absolute ValueUsing Absolute Value

California Standards:Number Sense 2.5Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers.

Algebra and Functions 1.1Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A).

What it means for you:You’ll use absolute value to find the difference between two numbers and see how absolute values apply to real-life situations.

Key words:• absolute value• comparison• difference

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Using Absolute ValueUsing Absolute ValueLesson

2.2.2

You use absolute value a lot in real life — often without even thinking about it.

This Lesson looks at some of the ways that absolute value can apply to everyday situations.

For example, if the temperature falls from 3 °C to –3 °C you might use it to find the overall change.

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Absolute Values Help Find Distances Between Numbers

Lesson

2.2.2

To find the distance between two numbers on the number line you could count the number of spaces between them.

Using subtraction is a quicker way — just subtract the lesser number from the greater.

If you did the subtraction the other way around you’d get a negative number — and distances can’t be negative.

8 – 6 = 2, so 8 and 6 are 2 units apart.

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But if you use absolute value bars you can do the subtraction in either order and you’ll always get a positive value for the distance.

Using Absolute ValueUsing Absolute ValueLesson

2.2.2

|6 – 8| = |–2| = 2 and |8 – 6| = 2

This is particularly useful when you’re finding the distance between a positive and negative number.

For any numbers a and b:The distance between a and b on the number line is |a – b|.

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Example 1

Solution follows…

Lesson

2.2.2

What is the distance between –3 and 5?

Solution

The distance between –3 and 5 is | –3 – 5 | = | –8 | = 8.

0 1 2 3–1–2–3–4–5–6 4 5 6

Distance of 8

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Guided Practice

Solution follows…

Lesson

2.2.2

Find the distance between the numbers given in each of Exercises 1–8.1. 1, 5 2. –3, –8

3. 6, –9 4. –1, 10

5. 3, –5 6. 5, –1

7. –1.2, 2.3 8. –0.3, 2.7

9. At 1 p.m., Amanda was 8 miles east of her home. She then traveled in a straight line west until she was 6 miles west of her home. How many miles did she travel?

4 5

15 11

8 6

3.5 3

14 miles

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Absolute Values are Used to Compare Things

Lesson

2.2.2

You can use absolute values to compare numbers when it doesn’t matter which side of a fixed point they are.

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Example 2

Solution follows…

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2.2.2

Find how far point A is above point B.

Solution

You’d get the same answer if you did the subtraction the other way around: |–35 m – 50 m| = |–85 m| = 85 m.

It doesn’t matter that B is below sea level and A is above. It’s the distance between them that’s important.

You can find this using: |50 m – (–35 m)| = |85 m| = 85 m.

Point A = 50 m (above sea level)

Point B = –35 m (below sea level)

Sea level = 0 m

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Guided Practice

Solution follows…

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2.2.2

10. A miner digs the shaft shown. What distance was he from the top of the crane when he finished digging?

11. The top of Mount Whitney is 14,505 ft above sea level. The bottom of Death Valley is 282 ft below sea level. How much higher is the top of Mount Whitney than the bottom of Death Valley?

20 ft

0 ft–10 ft–20 ft–30 ft–40 ft–50 ft

|–50 – 20| = |–70| = 70 ft

|14,505 – (–282)| = |14,787| = 14,787 ft

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Absolute Values Can Describe Limits

Lesson

2.2.2

Another use of absolute values is to describe the acceptable limits of a measurement.

It might not be important whether something is above or below a set value, but how far above or below it is.

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Example 3

Solution follows…

Lesson

2.2.2

The average temperature of the human body is 98.6 °F, but in a healthy person it can be up to 1.4 °F higher or lower.

The difference between a person’s temperature, x, and the average healthy temperature can be found using |98.6 – x|.

Aaron is feeling unwell so measures his temperature. It is 100.2 °F. Is Aaron’s temperature within the healthy range?

Solution

The difference between Aaron’s temperature and the average healthy temperature is |98.6 – 100.2| = |–1.6| = 1.6 °F.

Aaron’s temperature is outside of the normal healthy range.

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Wheel of diameter 31 inches: |30 – 31| = |–1| = 1 inch.


Example 4

Solution follows…

Lesson

2.2.2

A factory manufactures wheels that it advertises as no more than 1 inch away from 30 inches in diameter.

They use the expression |30 – d| to test whether wheels are within the advertised size (where d is the diameter).

Apply the expression to wheels of diameters 31, 29, and 35 inches, and say whether they meet the advertised standard.

Solution

This wheel is within the standard.

Wheel of diameter 29 inches: |30 – 29| = |1| = 1 inch. This wheel is within the standard.

Wheel of diameter 35 inches: |30 – 35| = |–5| = 5 inches.This wheel is not within the standard.

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Guided Practice

Solution follows…

Lesson

2.2.2

12. The height of a cupboard door should be no more than 0.05 cm away from 40 cm. The expression |40 – h| is used to check whether a door of height h cm fits the size requirement. If a door measures 40.049 cm, is it within the correct range?

13. Ms. Valesquez’s car needs a tire pressure, p, of 30 psi. It should be within 0.5 psi of the recommended value. She uses the expression |30 – p| to test whether the pressure is acceptable. Is a pressure of 29.4 psi acceptable?

|40 – 40.049| = 0.049. It is within the correct range.

|30 – 29.4| = 0.6. This isn’t an acceptable pressure.

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Independent Practice

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Lesson

2.2.2

Find the distance between the numbers given in each of Exercises 1–4.1. 9, –9 2. 3, –4

3. 5, 6 4. –32, –52

5. The table below shows the temperature at different times of day. How much did the temperature change by between 7 a.m. and 8 a.m.?

1 °C–5 °CTemperature

8 a.m.7 a.m.Time

18 7

1 20

6 °C

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Independent Practice

Solution follows…

Lesson

2.2.2

6. A person stands on a pier fishing. The top of their rod is 20 feet above sea level. The line goes vertically down and hooks a fish 13 feet below sea level. How long is the line?

7. Priscilla tries to keep the balance of her checking account, b, always less than $50 away from $200. She uses the expression |200 – b| to check that it is within these limits. Is a balance of $242.50 acceptable?

33 feet

|200 – 242.50| = 42.50. This is an acceptable balance.

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Round UpRound Up

Absolute values are used to find distances between numbers.

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2.2.2

In these situations, it doesn’t matter if the numbers are above or below the set point — it’s how far away they are that’s important.

They’re also useful when measurements are only allowed to be a certain distance away from a set value.

Documents

1 Lesson 2.2.2 Using Absolute Value. 2 Lesson 2.2.2 Using Absolute Value California Standards: Number Sense 2.5 Understand the meaning of the absolute