The Absolute Value Equation

Transcription

1 The Absolute Value Equation The absolute value of number is its distance from zero on the number line. The notation for absolute value is the presence of two vertical lines. 5 This is asking for the absolute value of 5 or the distance of 5 from zero on the number line. Five is five units from zero. So, the answer is 5. 5 = 5 There is also another location that gives a distance of 5. That location is at = 5 Every distance, except for zero, can be represented by two locations. For instance, - and both give distances of, -9 and 9 both give distances of 9, -3.4 and 3.4 both give distances of 3.4, and only 0 gives a distance of 0. ± = ± 9 = 9 ± 3.4 = = 0 The typical absolute value equation is asking for the location(s) which gives an identified distance. For instance, x = 4 asks the question: What value(s) will give a distance of 4? Since both -4 and 4 qualify as solutions, we can rewrite the absolute value equation without the absolute value notation by writing two equations. x = 4 x = 4 or x = 4 The two values (locations) that have a distance from zero of 4 are -4 and 4. This same technique can be used when the unknown is more involved. x + 8 = 10 asks the question: What value(s) will give a distance of 10? This is a little more difficult to see than the previous equation. Using the same technique as above, we can set the expression inside the absolute value symbol equal to -10 and 10. This will allow us to find the values of x that make the equation true. x + 8 = = 10 or x + 8= = 18 x = 18 = x = x = 9 x = 1 The two values (locations) that have a distance from zero of 10 are -9 and 1.

2 In order to use the previous technique for solving, you must isolate the absolute value symbol in the equation. You must do this to identify the distance in the problem. For instance, x + 5= 1 In this equation, the distance is not 1. The distance can be discovered only by isolating the absolute value. x + 5= x = 7 The distance of this problem is actually 7. So, we would then proceed to set the inside of the absolute value equal to -7 and 7 Use the above technique to solve the following absolute value equations. 1. x = 8. 1 = m 3. x 6= 1 4. a 9= 0 5. y 3 = c + 6 = a 3 5= r 6 = 11 answers: 1) x = 8 or x = 8 ) m = 1 or m = 1 3) x = 7 or x = 7 4) a = 9 or a = 9 5) y = 1 or y = 18 6) c = 8 or c = 4 7) a = 4 5 or a = 8) r = 1 or r = 7

3 The Absolute Value Inequality In order to understand the question being asked in the absolute value inequality, you must know well the question being asked in the absolute value equation. It s all about the distance, baby! Look at: x 5 The question here is: What value(s) of x will give a distance smaller than or equal to 5? There are many numbers that satisfy this inequality. First, identify the two locations that give a distance of 5. Those locations would be -5 and 5. We also want all locations that give distances smaller than 5. [ All of the locations between -5 and 5, including -5 and 5, give distances that satisfy the distance in question. So, we can say that we are interested in all values between and equal to -5 and 5. x 5 5 x 5 inequality notation [-5,5] interval notation Even if the difficulty of the problem is increased, we would be interested in the locations that give us the desired distance This is more involved than the first absolute value inequality. Since the absolute value symbol is isolated, we know that the distance of interest is 8. We want values that give this distance and distances that are smaller. Again, we find ourselves between and including -8 and x inequality notation [-6,] interval notation ]

4 Since distance is the key, you must always have the absolute value notation isolated in order to reveal the distance in question. x 5< 4 In this case, we are not interested in distances that are less than 4. Instead, we are interested in the distance that develops from isolating the absolute value symbol. x 5< x < 9 We want distances that are smaller than 9. The locations from -9 to 9, excluding -9 and 9, give the distances we are interested in finding. x < 9 9 < x < 9 inequality notation (-9,9) interval notation Use the above technique to solve the following absolute value inequalities. Give the inequality and interval notation for each. 1. y 4. a < c 4< 4. x d + 6 < 6. x y + 4 < x answers: 1) 4 y 4, [-4,4] ) 11 < a < 11, (-11,11) 3) 6 < c < 6, (-6,6) 4) x = 0 5) 4< d <, (-4,-) 6) 14 x 6, [-14,6] 7) 7 < y < 3, (-7,3) 8) 4 x 0, [-4,0]

5 Look at: x 3 The question here is: What value(s) of x will give a distance greater than or equal to 3? There are many numbers that satisfy this inequality. First, identify the two locations that give a distance of 3. Those locations would be -3 and 3. We also want all locations that give distances larger than 3. ] [ All of the values to the left of -3 and to the right of 3 also give distances that satisfy the distance in question. So, we can say that we are interested in all values less than equal to -3 or greater than equal to 3. x 3 x 3 or x 3 inequality notation (, 3] [3, interval notation Even if the difficulty of the problem is increased, we would be interested in the locations that give us the desired distance This is more involved than the first absolute value inequality. Since the absolute value symbol is isolated, we know that the distance of interest is 1. We want values that give this distance and distances that are greater. Again, we find ourselves less than equal to -1 or greater than equal to or x 8 or x 4 inequality notation (, 8] [8, interval notation Since distance is the key, you must always have the absolute value notation isolated in order to reveal the distance in question. x 9>4 In this case, we are not interested in distances that are greater than 4. Instead, we are interested in the distance that develops from isolating the absolute value symbol. x 9> x > 13 We want distances that are larger than 13. The locations to the left of -13 or to the right of 13, excluding -13 and 13, give the distances we are interested in finding. x > 13 x < 13 or x > 13 inequality notation (, 13) (13, interval notation

6 Use the above technique to solve the following absolute value inequalities. Give the inequality and interval notation for each. 1. y 4. a > c 4> 4. x d + 6 > 6. x y + 4 > x answers: 1) y 4 or y 4, ) a < 11 or a > 11 (, 4] [4, (, 11) (11, 3) c < 6 or c > 6 4) x : all real numbers (, 6) (6, (, 5) d < 4 or d > 6) x 14 or x 6 (, 4) (, (, 14] [6, 7) y < 7 or y > 3 8) x 4 or x 0 (, 7) (3, (, 4] [0,