Lesson 12: Solving and Graphing Absolute Value Equations. Representation of Absolute Value

Transcription

1 Lesson 12: Solving and Graphing Absolute Value Equations Definition of Absolute Value The absolute value of a number is The absolute value of a number is always Representation of Absolute Value Example 1: Evaluate an Expression with Absolute Value **Absolute value bars act as. Perform all operations inside the absolute value bars first. REMEMBER: x = x and -x = x Example 2: Solve the equation for x. Then check your solutions. Evaluate: 2 3a if a = -5 x = 9

2 Example 3: Solve the equation for x. Then check your solutions. y + 6 = 9 Example 4: Solve the equation for x. Then check your solutions. ½y -3-5 = 9 Check Your Answers: Check Your Answers: Example 5 2x = 0 Example 6 A chicken that you are cooking must reach an internal temperature of 165. A typical meat thermometer is accurate within plus or minus 2.5. Write an equation to determine the least and greatest temperatures of the meat. Solve the equation. To what temperature should you bake the chicken to ensure that the minimal temperature is reached?

3 Lesson 12: Solving and Graphing Absolute Value Equations Practice Problems Part 1: Decide whether the given number is a solution to the equation. (Hint: substitute the number into the equation. If the equation represents a true statement, then the number is a solution.) 1. 2x-8 = 2 where x = y = 30 where y = /3x 4 = 4/3 where x = x +4-6 = 30 where x = ½ 3-4y + 7 = 25 where y = -9 Part 2: Rewrite the absolute value equation as two linear equations. (This step will get you ready for solving the equations) 1. 4x 7 = x + 4 = /3y = /4x 8 = 30 Part 3: Solve the following equations and check your answers. 1. 3x 10 = /2s 12 = x + 3 = x + 4 = r 5-12 = x = /4y 9-8 = -7/8 8..5x = x 10 = x = 7.5 Part 4: Write an absolute value equation for each problem. Then solve the equation. 1. In order to brew coffee, you must have a brewing temperature of 195 F, plus or minus 5 degrees. Write an equation that could be used to find the minimum and maximum temperatures of a cup of coffee freshly brewed. Find the maximum and minimum temperatures for brewing coffee. 2. The typical length of a high school gymnasium is 90 feet, give or take 10 feet. Write an equation that could be used to find the minimum and maximum lengths of a high school gymnasium. Then find the minimum length of a gymnasium.

4 3. A box of cereal weighing 16 ounces, may not exactly weigh 16 ounces. The weight of a box of cereal can vary plus, or minus.4 ounces. Write an equation that could be used to determine the minimum and maximum weights of a box of cereal. If you bought 3 boxes of cereal, what is the total maximum weight of all three boxes together? 1. Decide whether the two answers given are a solution to the equation. Explain your answer. (3 points) 9 3x + 5 = 38 Solution set: {-8, -14} 2. Solve the following equations. Check your answers. (3 points each) 1. 2/3x 9 = x = Normal body temperature is 98.6 F. Your body temperature can change by as much a 1 F throughout the day. (3 points) Write an equation that can be used to determine the minimum and maximum normal body temperature of a person on any given day. Find the maximum normal body temperature of a person.

5 Lesson 12: Solving and Graphing Absolute Value Equations Practice Problems Answer Key Part 1: Decide whether the given number is a solution to the equation. (Hint: substitute the number into the equation. If the equation represents a true statement, then the number is a solution.) 1. 2x-8 = 2 where x = y = 30 where y =-7 2(-5)-8 = 2 Substitute -5 for x. -18 = 2 Evaluate inside of abs 18 = 2-18 = (-7) = 30 Substitute -7 for y. 34 = 30 Evaluate inside of abs = Since the two sides are not equal, we can conclude that -5 is not a solution to this equation. Since the two sides are not equal, we can conclude that -7 is not a solution to this equation. 3. 2/3x 4 = 4/3 where x = x +4-6 = 30 where x = -2 2/3(8) - 4 = 4/3 Substitute 8 for x. 4/3 = 4/3 Evaluate inside of abs 4/3 = 4/3 4/3 = 4/3 3 8(-2)+4-6=30 Substitute -2 for x = 30 Evaluate inside of abs 3(12) 6 = = = 30 3(12) -6 = 30 Since the two sides are equal, we can conclude that 8 is a solution to this equation. Since the two sides are equal, we can conclude that -2 is a solution to this equation 5. ½ 3-4y + 7 = 25 where y = -9 ½ 3 4(-9) + 7 = 25 Substitute -9 for y ½ = 25 Evaluate inside of abs ½(39) + 7 = = = 25 ½(39) + 7 = Since the two sides are not equal, we can conclude that -9 is not a solution to this equation.

6 Part 2: Rewrite the absolute value equation as two linear equations. (This step will get you ready for solving the equations) 1. 4x 7 = x + 4 = 16 The answer inside the absolute value sign can be positive 9 or negative 9. So, let one equation equal 9 and the other equation equal -9. 4x-7 = 9 4x 7 = -9 Only difference is one answer is 9 and the other -9. The answer inside the absolute value sign can be positive 16 or negative 16. So, let one equation equal 16 and the other equation equal x +4 =16-3x +4 = /3y = /4x 8 = 30 side of the equal sign by adding 10 to both sides. 2/3y = Add 10 2/3y+8 = = 30 The answer inside the absolute value sign can be positive 30 or negative 30. So, let one equation equal 30 and the other equation equal /3y+8 = 30 2/3y +8 = -30 side of the equal sign by subtracting 6 from both sides /4x 8 = 30 6 Subtract 6 1/4x 8 = = 24 The answer inside the absolute value sign can be positive 24 or negative 24. So, let one equation equal 24 and the other equation equal /4x-8 = 24 1/4x-8 = -24 Part 3: Solve the following equations and check your answers (using a calculator if possible). 1. 3x 10 = /2s 12 = 5 Step 1: Since the absolute value is isolated on the left side, we can write two equations. Step 1: Since the absolute value is isolated on the left side, we can write two equations. 3x-10 = 34 3x 10 = -34 1/2s - 12 = 5 1/2s - 12 = -5 3x-10 = 34 3x = Add 10 3x = 44 3x/3 = 44/3 Divide by 3 x = 44/3 3x 10 = -34 3x = Add 10 3x = -24 3x/3 = -24/3 Divide by 3 x = -8 Solution Set: {44/3, -8} 1/2s - 12 = 5 1/2s = 5+12 Add 12 1/2s = 17 (2)1/2s = 17(2) Multiply by 2 s = 34 1/2s - 12 = -5 1/2s = Add 12 1/2s = 7 (2)1/2s = 7(2) Multiply by 2 s = 14 Solution Set: {14, 34 }

7 3. -4x + 3 = x + 4 = 13 Step 1: Since the absolute value is isolated on the left side, we can write two equations. Algebra 1 Step 1: Since the absolute value is isolated on the left side, we can write two equations. -4x +3 = 57-4x+3 = -57-4x+3 = 57-4x +3 3 = 57 3 Subtract 3-4x = 54-4x/-4 = 54/-4 Divide by -4 x = -27/2 or x+3 = -57-4x +3 3 = Subtract 3-4x = -60-4x/-4 = -60/-4 Divide by -4 x = 15 Solution Set: {-13.5, 15}.45x +4 = 13.45x +4 = x +4 = 13.45x +4 4 = 13 4 Subtract 4.45x = 9.45x/.45 = 9/.45 Divide by.45 x = 20.45x +4 = x +4 4 = Subtract 4.45x = x/.45 = -17/.45 Divide by.45 x = Solution Set: {20, 37.78} 5. 8r 5-12 = x = 3 side by adding 12 to both sides. 8r = side by subtracting 10 from both sides. 5x = r 5 = 29 New Equation 5x-3 = -7 New Equation 8r 5 = 29 8r 5 = -29 8r 5 = 29 8r = Add 5 8r = 34 8r/8 = 34/8 Divide by 8 r = 17/4 or r 5 = -29 8r = Add 5 8r = -24 8r/8 = -24/8 Divide by 8 r =-3 Stop Here!!! And think Can any expression inside of an absolute value sign equal a negative number? NO, the absolute value of any expression must be positive; therefore, we don t need to go any further. The solution set is: the empty set: {Ø}. Solution Set: {-3, 17/4}

8 7. 3/4y 9-8 = -7/8 8..5x = 8.25 side by adding 8 to both sides. side by subtracting 4 from both sides. 3/4y = -7/8 +8 3/4y 9 = 57/8 New Equation.5x = x + 4 = 4.25 New Equation 3/4y 9 = 57/8 3/4y 9 = -57/8 3/4y 9 = 57/8 3/4y = 57/8 + 9 Add 9 3/4y = 129/8 (4/3)3/4y = 129/8(4/3) Multiply by 4/3 y = 43/2 or /4y 9 = -57/8 3/4y = -57/8 + 9 Add 9 3/4y = 15/8 (4/3)3/4y = 15/8(4/3) Multiply by 4/3 y =5/2 or 2.5 Solution Set: {5/2, 43/2}.5x + 4= x + 4= x + 4 = x = Subtract 4.5x =.25.5x/.5 =.25/.5 Divide by.5 x=.5 or 1/2.5x + 4 = x = Subtract 4.5x = x/.5 = -8.25/.5 Divide by.5 x= Solution Set: {-16.5, 0.5} x 10 = x = 7.5 side by subtracting 18 from both sides x 10 = x 10 = 6 New Equation.25x 10 = 6.25x 10 = -6.25x 10 = 6.25x = Add 10.25x = 16.25x/.25 = 16/.25 Divide by.25 x = 64.25x 10 = -6.25x = Add 10.25x = 4.25x/.25 = 4/.25 Divide by.25 x = 16 side by adding 12 to both sides. 4x = x 2.5 = 19.5 New Equation 4x 2.5 = x 2.5 = x 2.5 = x = Add 2.5 4x = 22 4x/4 = 22/4 Divide by 4 x = 11/2 or 5.5 4x 2.5 = x = Add 2.5 4x = -17 4x/4 = -17/4 Divide by 4 x = -17/4 or Solution Set: {16, 64} Solution Set: {-17/4, 11/2} or {-4.25, 5.5}

9 Part 4: Write an absolute value equation for each problem. Then solve the equation. 1. In order to brew coffee, you must have a brewing temperature of 195 F, plus or minus 5 degrees. Write an equation that could be used to find the minimum and maximum temperatures of a cup of coffee freshly brewed. In this problem we know that the difference between the actual temperature and 195 must be with plus or minus 5 degrees. Therefore, we know our variable 195 must be within the absolute value sign, and that value must equal 5 degrees. Let t = the brewing temperature of the coffee. t 195 = 5 is the equation we can use to find the minimum and maximum temperatures. Find the maximum and minimum temperatures for brewing coffee. We can solve this equation to find the minimum and maximum temperatures of the coffee. Let s solve for t. t 195 = 5 t 195 = 5 t 195 = -5 Write two linear equations. t = t = Add 195 to both equations t = 200 t = 190 The minimum temperature for brewing coffee is 190 degrees and the maximum temperature for brewing coffee is 200 degrees. 2. The typical length of a high school gymnasium is 90 feet, give or take 10 feet. Write an equation that could be used to find the minimum and maximum lengths of a high school gymnasium. Then find the minimum length of a gymnasium. In this problem, we know that difference between the length of the gymnasium and 90 feet must be plus or minus 10 feet. Let x = the actual length of a gymnasium. x 90 = 10 is the equation we can use to find the minimum or maximum length of the gymnasium. Now, we can solve to find minimum length of the gymnasium. x-90 = 10 x 90 = 10 x 90 = -10 Write two linear equations. x = x = Add 90 to both sides x = 100 x = 80 The minimum length of a gymnasium is 80 feet.

10 3. A box of cereal weighing 16 ounces, may not exactly weigh 16 ounces. The weight of a box of cereal can vary plus, or minus.4 ounces. Write an equation that could be used to determine the minimum and maximum weights of a box of cereal. In this problem, we know that the difference between the actual weight of a box of cereal and 16 is plus or minus.4 ounces. Let x = the actual weight of the cereal. x 16 =.4 is the equation that we can use to determine the minimum and maximum weights of the cereal. If you bought 3 boxes of cereal, what is the total maximum weight of all three boxes together? Using the equation from above: x- 16 =.4 x 16 =.4 x 16 = -.4 Write two linear equations x = x = Add 16 x = 16.4 x = 15.6 The maximum weight of one box is 16.4 ounces. Therefore, the maximum weight of 3 boxes of cereal would be 49.2 ounces (16.4 * 3 = 49.2) 1. Decide whether the two answers given are a solution to the equation. Explain your answer. (3 points) 9 3x + 5 = 38 Solution set: {-8, -14} In order to decide whether the given answers are solution to the equation, we must substitute each solution into the original equation. If the expression is true, then the solution set is correct. 9 3x + 5 = 38 Solution set: {-8, -14} 9 3(-8) + 5 = 38 Substitute -8 for x = = = 38 This is a true statement 9 3x + 5 = (-14) + 5 = 38 Substitute = = Since 56 does not equal 38, -14 is not a solution. Therefore, the solution set is incorrect. -8 is a solution, but -14 is not a solution.

11 2. Solve the following equations. Check your answers. (3 points each) 1. 2/3x 9 = x = 29 Step 1: Since the absolute value is isolated on the left side, we can write two equations. 2/3x 9 = 21 2/3x 9 = -21 2/3x 9 = 21 2/3x 9 +9 = Add 9 2/3x = 30 (3/2)2/3x = 30(3/2) Multiply by 3/2 x = 45 2/3x 9 = -21 2/3x 9 +9 = Add 9 2/3x =-12 (3/2)2/3x = -12(3/2) Multiply by 3/2 x = -18 Solution Set: {-18, 45} side by subtracting 12 from both sides x = x = 17 New Equation 9x = 17 9x = -17 9x = 17 9x = Subtract 5.5 9x = x / 9 = 11.5 / 9 Divide by 9 x = 23/18 or x = -17 9x = Subtract 5.5 9x = x / 9 = / 9 Divide by 9 x = -2.5 Solution Set: {-2.5, 1.28} 3. Normal body temperature is 98.6 F. Your body temperature can change by as much a 1 F throughout the day. (3 points) Write an equation that can be used to determine the minimum and maximum normal body temperature of a person on any given day. We know that the difference in actual body temperature and 98.6 degrees is plus or minus 1 degree. Let x = actual body temperature. x 98.6 = 1 is the equation used to determine the minimum and maximum body temperature. Find the maximum normal body temperature of a person. x 98.6 = 1 Given equation x 98.6 = 1 x 98.6 = -1 Write two linear equations x = x = Add 98.6 x = 99.6 x = The minimum normal body temperature of a person is 97.6 F and the maximum normal body temperature is 99.6 F.