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

2. 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. 01234-2-3-4 x |-3| = 3 units |2| = 2 units Examples: Draw a diagram to represents | -3 | and | 2 | on the same diagram. 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. Examples: Solve the following: a) | 2x - 5 | = x + 23 - (2x - 5) = x + 23+ ( 2x - 5 ) = x + 22 -2x + 5 = x + 23 -3x + 5 = 23 -3x = 18 x = -6 2x - 5 = x + 22 x - 5 = 22 x = 27 Solve 2x - 5= 0 x = 2.5 (2x - 5) is negative for x < 2.5 (2x - 5) is positive for x ≥ 2.5 Both solution are allowed, hence x = -6 or x = 27. [looking for x < 2.5][looking for x ≥ 2.5] b) | 5 - 4x | = 2x - 7 Solve 5 - 4x= 0 x = 1.25 (5 - 4x) is negative for x > 1.25 (5 - 4x) is positive for x ≤ 1.25 [looking for x > 1.25][looking for x ≤ 1.25] - ( 5 - 4x ) = 2x - 7+ ( 5 - 4x ) = 2x - 7 - 5 + 4x = 2x - 7 - 5 + 2x = - 7 2x = - 2 x = - 1 5 - 4x = 2x - 7 5 - 6x = - 7 - 6x = - 12 x = 2 Both answer are NOT allowed. The equation has no solution. But x must be ≤ 1.25 But x must be > 1.25 Test your answers by substitution.

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

6. Inequality Examples. Solve and graph number line solution for the following: a) | 2x + 5 | < 17 Solve 2x + 5= 0 x = -2.5 (2x + 5) is negative for x ≥ -2.5 (2x + 5) is positive for x < -2.5 - ( 2x + 5) < 17+ ( 2x + 5 ) < 17 [Negative sol.][Positive sol.] 2x + 5 > -17 2x > -22 x > -11 2x + 5 < 17 2x < 12 x < 6 (allowed) -1106 x Solution: -11 < x < 6 b) | 5x + 1 | ≥ 3x +15 Solve 5x + 1= 0 x = - 0.2 (5x + 1) is negative for x < -0.2 (5x + 1) is positive for x ≥ -0.2 [Negative sol.][Positive sol.] - ( 5x + 1 ) ≥ 3x +15 - 5x - 1 ≥ 3x +15 - 8x - 1 ≥ 15 - 8x ≥ 16 x ≤ -2 + ( 5x + 1 ) ≥ 3x +15 5x + 1 ≥ 3x +15 2x + 1 ≥ 15 2x ≥ 14 x ≥ 7 Solution: x ≤ -2 OR x ≥ 7 -207 x (allowed)

7. c) | 5 - 4x | < 2x - 7 Solve 5 - 4x= 0 x = 1.25 (5 - 4x) is negative for x > 1.25 (5 - 4x) is positive for x ≤ 1.25 [looking for x > 1.25][looking for x ≤ 1.25] - ( 5 - 4x ) < 2x - 7+ ( 5 - 4x ) < 2x - 7 - 5 + 4x < 2x - 7 - 5 + 2x < - 7 2x < - 2 x < - 1 5 - 4x < 2x - 7 5 - 6x < - 7 - 6x < - 12 x > 2 Both answer are NOT allowed. The equation has no solution. But x must be ≤ 1.25 But x must be > 1.25 Test your answers by substitution. d) | 12 - x | ≤ x Solve 12 - x= 0 x = 12 (5 - 4x) is negative for x > 12 (5 - 4x) is positive for x ≤ 12 [looking for x > 125][looking for x ≤ 12] - ( 12 - x ) ≤ x x - 12 ≤ x - 12 ≤ 0 A true statement, but not an answer! + ( 12 - x ) ≤ x 12 - x ≤ x - 2x ≤ -12 x ≥ 6 (allowed) (NOT allowed) Solution: x ≥ 6 -1206 x

8. Exercise: Solve and graph number line solution for the following: a) | 4x + 6| ≥ 14x ≤ -5 or x ≥ 2 b) | 2x - 5| < x + 10x > 1 2 / 3 or x < 15 c) | 20 - x| > 2x + 5x < 5 or x < -25 (not allowed) d) | x + 5| ≤ 2xx ≥ 5 or x ≥ - 5 / 3 (not allowed) 12/312/3 015 x -2505 x -502 x -5/3-5/3 06 x