1-2 Adding and Subtracting Real Numbers Warm Up Lesson Presentation Holt Algebra 1 Warm Up Lesson Presentation Lesson Quiz

Warm Up Simplify. 1. |–3| 3 2. –|4| –4 Write an improper fraction to represent each mixed number. 2 14 6 55 3. 4 4. 7 3 3 7 7 Write a mixed number to represent each improper fraction. 2 3 12 2 5 24 5. 6. 5 9

Objectives Add real numbers. Subtract real numbers.

Vocabulary absolute value opposites additive inverse

All the numbers on a number line are called real numbers. You can use a number line to model addition and subtraction of real numbers. Addition To model addition of a positive number, move right. To model addition of a negative number move left. Subtraction To model subtraction of a positive number, move left. To model subtraction of a negative number, move right.

Example 1A: Adding and Subtracting Numbers on a Number Line Add or subtract using a number line. –4 + (–7) Start at 0. Move left to –4. To add –7, move left 7 units. + (–7) –4 11 10 9 8 7 6 5 4 3 2 1 –4+ (–7) = –11

Example 1B: Adding and Subtracting Numbers on a Number Line Add or subtract using a number line. 3 – (–6) Start at 0. Move right to 3. To subtract –6, move right 6 units. –6 + 3 -3 -2 -1 1 2 3 4 5 6 7 8 9 3 – (–6) = 9

-3 -2 -1 1 2 3 4 5 6 7 8 9 Check It Out! Example 1a Add or subtract using a number line. –3 + 7 Start at 0. Move left to –3. To add 7, move right 7 units. +7 –3 -3 -2 -1 1 2 3 4 5 6 7 8 9 –3 + 7 = 4

Check It Out! Example 1b Add or subtract using a number line. –3 – 7 Start at 0. Move left to –3. To subtract 7 move left 7 units. –3 –7 11 10 9 8 7 6 5 4 3 2 1 –3 – 7 = –10

Check It Out! Example 1c Add or subtract using a number line. –5 – (–6.5) Start at 0. Move left to –5. To subtract negative 6.5 move right 6.5 units. –5 – (–6.5) 8 7 6 5 4 3 2 1 1 2 –5 – (–6.5) = 1.5

The absolute value of a number is the distance from zero on a number line. The absolute value of 5 is written as |5|. 5 units 5 units - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 |–5| = 5 |5| = 5

Example 2A: Adding Real Numbers When the signs of numbers are different, find the difference of the absolute values: Use the sign of the number with the greater absolute value. The sum is negative.

Example 2B: Adding Real Numbers y + (–2) for y = –6 y + (–2) = (–6) + (–2) First substitute –6 for y. When the signs are the same, find the sum of the absolute values: 6 + 2 = 8. (–6) + (–2) –8 Both numbers are negative, so the sum is negative.

Check It Out! Example 2a Add. –5 + (–7) –5 + (–7) = 5 + 7 When the signs are the same, find the sum of the absolute values. 5 + 7 = 12 Both numbers are negative, so the sum is negative. –12

Check It Out! Example 2b Add. –13.5 + (–22.3) –13.5 + (–22.3) When the signs are the same, find the sum of the absolute values. 13.5 + 22.3 –35.8 Both numbers are negative so, the sum is negative.

Check It Out! Example 2c Add. x + (–68) for x = 52 First substitute 52 for x. x + (–68) = 52 + (–68) When the signs of the numbers are different, find the difference of the absolute values. 68 – 52 Use the sign of the number with the greater absolute value. –16 The sum is negative.

Two numbers are opposites if their sum is 0 Two numbers are opposites if their sum is 0. A number and its opposite are on opposite sides of zero on a number line, but are the same distance from zero. They have the same absolute value.

Additive inverses 11 – 6 = 5 11 + (–6) = 5 A number and its opposite are additive inverses. To subtract signed numbers, you can use additive inverses. Subtracting 6 is the same as adding the inverse of 6. Additive inverses 11 – 6 = 5 11 + (–6) = 5 Subtracting a number is the same as adding the opposite of the number.

Example 3A: Subtracting Real Numbers –6.7 – 4.1 –6.7 – 4.1 = –6.7 + (–4.1) To subtract 4.1, add –4.1. When the signs of the numbers are the same, find the sum of the absolute values: 6.7 + 4.1 = 10.8. = –10.8 Both numbers are negative, so the sum is negative.

Example 3B: Subtracting Real Numbers 5 – (–4) 5 − (–4) = 5 + 4 To subtract –4 add 4. 9 Find the sum of the absolute values.

Example 3C: Subtracting Real Numbers First substitute for z. To subtract , add . Rewrite with a denominator of 10.

Example 3C Continued When the signs of the numbers are the same, find the sum of the absolute values: . Write the answer in the simplest form. Both numbers are negative, so the sum is negative.

Check It Out! Example 3a Subtract. 13 – 21 13 – 21 = 13 + (–21) To subtract 21 add –21. When the signs of the numbers are different, find the difference of the absolute values: 21 – 13 = 8. –8 Use the sign of the number with the greater absolute value.

Check It Out! Example 3b Subtract. To subtract add . –3 1 2 3 When the signs of the numbers are the same, find the sum of the absolute values: = 4. 3 1 2 + 4 Both numbers are positive so, the sum is positive.

Check It Out! Example 3c Subtract. x – (–12) for x = –14 x – (–12) = –14 – (–12) First substitute –14 for x. –14 + (12) To subtract –12, add 12. When the signs of the numbers are different, find the difference of the absolute values: 14 – 12 = 2. Use the sign of the number with the greater absolute value. –2

elevation at top of iceberg Example 4: Oceanography Application An iceberg extends 75 feet above the sea. The bottom of the iceberg is at an elevation of –247 feet. What is the height of the iceberg? Find the difference in the elevations of the top of the iceberg and the bottom of the iceberg. elevation at top of iceberg 75 elevation at bottom of iceberg –247 Minus – 75 – (–247) 75 – (–247) = 75 + 247 To subtract –247, add 247. Find the sum of the absolute values. = 322 The height of the iceberg is 322 feet.

elevation at top of iceberg Check It Out! Example 4 What if…? The tallest known iceberg in the North Atlantic rose 550 feet above the oceans surface. How many feet would it be from the top of the tallest iceberg to the wreckage of the Titanic, which is at an elevation of –12,468 feet? elevation at top of iceberg 550 Minus elevation of the Titanic –12,468 – 550 – (–12,468) To subtract –12,468, add 12,468. 550 – (–12,468) = 550 + 12,468 Find the sum of the absolute values. = 13,018 Distance from the iceberg to the Titanic is 13,018 feet.

Lesson Quiz Add or subtract using a number line. 2. –5 – (–3) –2 1. –2 + 9 7 Add or subtract. 3. –23 + 42 19 4. 4.5 – (–3.7) 8.2 5. 6. The temperature at 6:00 A.M. was –23°F. At 3:00 P.M. it was 18°F. Find the difference in the temperatures. 41°F