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Revision of Absolute Equations

Revision of Absolute Equations. Mathematics Course ONLY (Does not include Ext 1 & 2) By I Porter. x. -4. -3. -2. -1. 0. 1. 2. 3. 4. Introduction. The absolute value of a number x in R , written | x |, is given by the non-negative number that defines its magnitude .

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Revision of Absolute Equations

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  1. Revision of Absolute Equations Mathematics Course ONLY (Does not include Ext 1 & 2) By I Porter

  2. x -4 -3 -2 -1 0 1 2 3 4 Introduction The absolute value of a number x in R, written | x |, is given by the non-negative number that defines its magnitude. Since x is a real number, it can be represented by a point on the number line. It is useful to regard | x | as the distance of a point x from the origin, 0, since distance is a measure by positive numbers, the | x | is positive for all x ≠ 0. Examples: Draw a diagram to represents | -3 | and | 2 | on the same diagram. |-3| = 3 units |2| = 2 units More generally, | x - y | may be considered as the distance between two points x and y On the same number line and hence | x - y | is a positive number if x > y or if x < y.

  3. [Introduction continued] More generally, | x - y | may be considered as the distance between two points x and y On the same number line and hence | x - y | is a positive number if x > y or if x < y. Two important theorem you should know. (a) | xy | = | x | . | y | (b) | x + y | ≤ | x | + | y | , (the triangle inequality) and | x + y | = | x | + | y | , when and only when x & y are either both are zero or both have the same sign. More simply put, the contents of | … | could be POSITIVE or NEGATIVE.

  4. Solve 5 - 4x = 0 x = 1.25 (5 - 4x) is negative for x > 1.25 (5 - 4x) is positive for x ≤ 1.25 Examples: Solve the following: Solve 2x - 5 = 0 x = 2.5 (2x - 5) is negative for x < 2.5 (2x - 5) is positive for x ≥ 2.5 | 5 - 4x | = 2x - 7 b) | 2x - 5 | = x + 23 a) [looking for x < 2.5] [looking for x ≥ 2.5] [looking for x > 1.25] [looking for x ≤ 1.25] - (2x - 5) = x + 23 + ( 2x - 5 ) = x + 22 - ( 5 - 4x ) = 2x - 7 + ( 5 - 4x ) = 2x - 7 2x - 5 = x + 22 - 5 + 4x = 2x - 7 5 - 4x = 2x - 7 -2x + 5 = x + 23 - 5 + 2x = - 7 5 - 6x = - 7 -3x + 5 = 23 x - 5 = 22 2x = - 2 - 6x = - 12 -3x = 18 x = 27 x = - 1 x = 2 x = -6 But x must be ≤ 1.25 But x must be > 1.25 Both solution are allowed, hence x = -6 or x = 27. Both answer are NOT allowed. The equation has no solution. Test your answers by substitution.

  5. There are 4 possible number of solutions: 0 (zero) solutions 1 (one) solution 2 (two) solutions ∞ (infinite) number of solutions. Exercise: Solve the following. a) | 4x + 1 | = 29 b) | 5x - 1 | = 3x - 15 x = 7 x = -7, not allowed Positive sol. x ≥ -1/4 Positive sol. x ≥ 1/5 x = -71/2 x = 2, not allowed Negative sol. x < -1/4 Negative sol. x < 1/5 Both are solutions. There is NO solution. c) | 12 - 3x | = 20 - x d) | 10 - x | = x x = -4 x = 5 Positive sol. x ≤ 4 Positive sol. x ≤ 10 x = 8 x = has no solution Negative sol. x > 4 Negative sol. x > 10 Both are solutions. There is a single solution x = 5.

  6. x x -11 0 6 -2 0 7 Inequality Examples. Solve and graph number line solution for the following: Solve 2x + 5 = 0 x = -2.5 (2x + 5) is negative for x ≥ -2.5 (2x + 5) is positive for x < -2.5 Solve 5x + 1 = 0 x = - 0.2 (5x + 1) is negative for x < -0.2 (5x + 1) is positive for x ≥ -0.2 a) | 2x + 5 | < 17 b) | 5x + 1 | ≥ 3x +15 [Negative sol.] [Positive sol.] [Negative sol.] [Positive sol.] - ( 5x + 1 ) ≥ 3x +15 + ( 5x + 1 ) ≥ 3x +15 - ( 2x + 5) < 17 + ( 2x + 5 ) < 17 2x + 5 > -17 2x + 5 < 17 - 5x - 1 ≥ 3x +15 5x + 1 ≥ 3x +15 2x > -22 2x < 12 - 8x - 1 ≥ 15 2x + 1 ≥ 15 x > -11 x < 6 - 8x ≥ 16 2x ≥ 14 (allowed) (allowed) x ≤ -2 x ≥ 7 (allowed) (allowed) Solution: -11 < x < 6 Solution: x ≤ -2 OR x ≥ 7

  7. x -12 0 6 Solve 5 - 4x = 0 x = 1.25 (5 - 4x) is negative for x > 1.25 (5 - 4x) is positive for x ≤ 1.25 Solve 12 - x = 0 x = 12 (5 - 4x) is negative for x > 12 (5 - 4x) is positive for x ≤ 12 d) | 12 - x | ≤ x | 5 - 4x | < 2x - 7 c) [looking for x > 1.25] [looking for x ≤ 1.25] [looking for x > 125] [looking for x ≤ 12] - ( 5 - 4x ) < 2x - 7 + ( 5 - 4x ) < 2x - 7 - ( 12 - x ) ≤ x + ( 12 - x ) ≤ x - 5 + 4x < 2x - 7 5 - 4x < 2x - 7 x - 12 ≤ x 12 - x ≤ x - 2x ≤ -12 - 5 + 2x < - 7 5 - 6x < - 7 - 12 ≤ 0 2x < - 2 - 6x < - 12 x ≥ 6 A true statement, but not an answer! x < - 1 x > 2 (allowed) But x must be ≤ 1.25 But x must be > 1.25 (NOT allowed) (NOT allowed) Solution: x ≥ 6 Both answer are NOT allowed. The equation has no solution. Test your answers by substitution.

  8. x -5 0 2 x 0 12/3 15 x -25 0 5 x -5/3 0 6 Solve and graph number line solution for the following: Exercise: a) | 4x + 6| ≥ 14 x ≤ -5 or x ≥ 2 b) | 2x - 5| < x + 10 x > 12/3 or x < 15 c) | 20 - x| > 2x + 5 x < 5 or x < -25 (not allowed) d) | x + 5| ≤ 2x x ≥ 5 or x ≥ -5/3(not allowed)

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