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**Solving Equations Containing Integers**

2-4 Solving Equations Containing Integers Pre-Algebra Warm Up Problem of the Day Lesson Presentation

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**2-4 Solving Equations Containing Integers Warm Up**

Pre-Algebra 2-4 Solving Equations Containing Integers Warm Up Add, subtract, multiply, or divide. 41 2. 23 – 19 4 3. 12 · 3 36 4. 6(–7) –42 –64 8 5. –8 6. –250 + (–85) –335

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**+ + = 15 + – = 3 + – = 5 + – = 7 Problem of the Day**

What are the numerical values of the , the , and the ? = 15 – = 3 – = 5 – = 7 = 6, = 4, = 5

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**Learn to solve equations with integers.**

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When you are solving equations with integers, your goal is the same as with whole numbers: isolate the variable on one side of the equation. + – + – + – Recall that the sum of a number and its opposite is 0. When you add the opposite to get 0, you can isolate the variable. 3 + (–3) = 0 a + (–a) = 0

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**Additional Example 1A & 1B: Adding and Subtracting to Solve Equations**

Add 3 to both sides. x – = –3 x = – 3 B. –5 + r = 9 –5 + r = 9 – r = 9 + 5 Add 5 to both sides. r = 14

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**The variable is already isolated. –6 + 8 = n 2 = n Add integers.**

Additional Example 1C & 1D: Adding and Subtracting to Solve Equations Continued Solve. C. –6 + 8 = n The variable is already isolated. –6 + 8 = n 2 = n Add integers. D. z + 6 = –3 z + 6 = –3 –6 Add –6 to each side. z = –9

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**Solve. A. B. p – 7 = – 9 p – 7 = – 9 p – 7 + 7 = – 9 + 7**

Try This: Example 1A & 1B Solve. A. p – 7 = – 9 p – 7 = – 9 p – = – 9 + 7 Add 7 to both sides. p – = –2 p = –2 B. –2 + g = 5 –2 + g = 5 – g = 5 + 2 Add 2 to both sides. g = 7

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**The variable is already isolated. –1 + 7 = r**

Try This: Example 1C & 1D Solve. C. –1 + 7 = r The variable is already isolated. –1 + 7 = r 6 = r Add integers. D. a + 9 = –9 a + 9 = –9 –9 Subtract 9 from each side. a = –18

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**Solve. A. –5x = 35 –5x = 35 –5 Divide both sides by –5. x = –7**

Additional Example 2A: Multiplying and Dividing to Solve Equations Continued Solve. A. –5x = 35 –5x = 35 –5 Divide both sides by –5. x = –7

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**Multiply both sides by –4.**

Additional Example 2B: Multiplying and Dividing to Solve Equations Continued Solve. z –4 B. = 5 – = –4 5 z –4 Multiply both sides by –4. z = –20

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**Solve. A. –7x = 42 –7x = 42 Divide both sides by –7. –7 x = –6**

Try This: Example 2A Solve. A. –7x = 42 –7x = 42 –7 Divide both sides by –7. x = –6

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**Multiply both sides by –3.**

Try This: Example 2 Solve. z –3 B. = 9 – = –3 9 z –3 Multiply both sides by –3. z = –27

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**Additional Example 3: Problem Solving Application**

Sarah heard on the morning news that the temperature had dropped 26 degrees since midnight. In the morning, the temperature was –8°F. What was the temperature at midnight?

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**Understand the Problem**

Additional Example 3 Continued 1 Understand the Problem The answer is the temperature at midnight. List the important information: The temperature dropped 26 degrees since midnight. In the morning it was –8°F. Show the relationship or the information: temperature at midnight morning temperature degrees that temperature dropped – =

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**Additional Example 3 Continued**

2 Make a Plan Write an equation and solve it. Let t represent the temperature at midnight and use the equation model. t – 26 = –8 Solve 3 t – 26 = –8 + 26 Add 26 to both sides. t = 18 The temperature at midnight was 18°F.

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**Additional Example 3 Continued**

4 Look Back The temperature at midnight was positive. Its value is less than the absolute value of the drop in temperature. This makes sense, since the morning temperature was negative.

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Try This: Example 3 Tim has two tug boats in a pulling contest. The boat on the left was pulling with a force of –23 newtons (N). If the net force is 5 N, what force is the boat on the right exerting on the rope?

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**Understand the Problem**

Try This: Example 3 Continued 1 Understand the Problem The answer is the force that the right tug boat is exerting. List the important information: The boat on the left pulls with a force of –23 N. The net force is 5 N. Show the relationship of the information: net force left boat’s force = right boat’s force +

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**Try This: Example 3 Continued**

2 Make a Plan Write an equation and solve it. Let f represent the force in newtons. 5 = –23 + f Solve 3 5 = –23 + f +23 +23 Add 23 to both sides. 28 = f The right boat is exerting a force of 28 newtons on the rope. –

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**Try This: Example 3 Continued**

Look Back 4 The left tug boat was pulling at a force of –23 N to the left. Since the total force on the tow rope was 5 N, the right tug boat was exerting a greater force in the positive direction. Thus the tug boat is moving to the right.

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Lesson Quiz: Part 1 Solve. 1. t + 9 = –8 t = –17 2. –15 = 3b b = –5 x –4 = –7 x = 28 4. z – 16 = 30 z = 46

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Lesson Quiz: Part 2 5. A roller coaster descends down a hill at a rate of 80 feet per second. The bottom of the hill is 400 feet from the top. How long will it take the coaster riders to reach the bottom? 5 seconds

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