Revision of Absolute Equations

Mathematics Course ONLY

(Does not include Ext 1 & 2)

By I Porter

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 measureby positive numbers, the | x | is positive for all x ≠ 0.

0 1 2 3 4-1-2-3-4x

|-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 yOn the same number line and hence | x - y | is a positive number if x > y or if x < y.

[Introduction continued]

More generally, | x - y | may be considered as the distance between two points x and yOn 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 eitherboth are zero or both have the same sign.

More simply put, the contents of | … | could be POSITIVE or NEGATIVE.

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

Solve2x - 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, hencex = -6 or x = 27.

[looking for x < 2.5] [looking for x ≥ 2.5]

b) | 5 - 4x | = 2x - 7

Solve5 - 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.

Exercise: Solve the following.

There are 4 possible number of solutions:0 (zero) solutions1 (one) solution2 (two) solutions∞ (infinite) number of solutions.

a) | 4x + 1 | = 29 b) | 5x - 1 | = 3x - 15

c) | 12 - 3x | = 20 - x d) | 10 - x | = x

Positive sol. x ≥ -1/4 x = 7

Negative sol. x < -1/4 x = -71/2

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.

Inequality Examples.Solve and graph number line solution for the following:

a) | 2x + 5 | < 17

Solve2x + 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) (allowed)

-11 0 6x

Solution: -11 < x < 6

b) | 5x + 1 | ≥ 3x +15

Solve5x + 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

-2 0 7x

(allowed) (allowed)

c) | 5 - 4x | < 2x - 7

Solve5 - 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

Solve12 - 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) (NOT allowed)

Solution: x ≥ 6

-12 0 6x

Exercise: Solve and graph number line solution for the following:

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)

12/30 15x

-25 0 5x

-5 0 2x

-5/3 0 6x