SOLVING ABSOLUTE VALUE EQUATIONS

The general form of an absolute value equation is

|ax + b|  =  k

(where a and b are real numbers, k ≥ 0)

Step 1 :

Rewrite the equation, remove the absolute value symbol, and divide it into two branches.

ax + b  =  k (or) ax + b  =  -k

Step 2 :

Solve for x from step 1.

Some properties of an absolute value equation are here  :

(i)  |ax + b|  =  |cx + d|

(ax + b)  =  (cx + d)   (or)  (ax + b)  =  -(cx + d)

(ii)  |[(ax + b)/(cx + d)]| = k

[(ax + b)/(cx + d)]  =  k (or) [(ax + b)/(cx + d)]  =  -k

(iii) |ax + b| < 0

There is no solution.

Write the solution set of each equation.

Problem 1 :

│x - 5│ = 12

Solution :

│x - 5│ = 12

x - 5 = 12 (or) x - 5 = -12

x = 17 (or) x = -7

Problem 2 :

│x + 8│ = 6

Solution :

│x + 8│ = 6

x + 8 = 6 (or) x + 8 = -6

x = -2 (or) x = -14

Problem 3 :

│2a - 5│ = 7

Solution :

│2a - 5│ = 7

2a - 5 = 7 (or) 2a - 5 = -7

2a = 12 (or) 2a = -2

a = 6 (or) a = -1

Problem 4 :

│5b - 10│ = 25

Solution :

│5b - 10│ = 25

5b - 10 = 25 (or) 5b - 10 = -25

5b = 35 (or) 5b = -15

b = 7 (or) b = -3

Problem 5 :

│3x - 12│ = 9

Solution :

│3x - 12│ = 9

3x - 12 = 9 (or) 3x - 12 = -9

3x = 21 (or) 3x = 3

x = 7 (or) x = 1

Problem 6 :

│4y + 2│ = 14

Solution :

│4y + 2│ = 14

4y + 2 = 14 (or) 4y + 2 = -14

4y = 12 (or) 4y = -16

y = 3 (or) y = -4

Problem 7 :

│35 - 5x│ = 10

Solution :

│35 - 5x│ = 10

35 - 5x = 10 (or) 35 - 5x = -10

-5x = -25 (or) -5x = -45

x = 5 (or) x = 9

Problem 8 :

│-5a│+ 7 = 22

Solution :

│-5a│+ 7 = 22

Subtract 7 on both sides,

│-5a│+ 7 - 7 = 22 - 7

│-5a│= 15

-5a = 15 (or) -5a = -15

a = -3 (or) a = 3

Problem 9 :

│8 + 2b│ - 3 = 9

Solution :

│8 + 2b│ - 3 = 9

│8 + 2b│ - 3 + 3 = 9 + 3

│8 + 2b│= 12

8 + 2b = 12 (or) 8 + 2b = -12  

2b = 4 (or) 2b = -20

b = 2 (or) b = -10

Problem 10 :

│2x - 5│ + 2 = 13

Solution:

│2x - 5│ + 2 = 13

│2x - 5│ + 2 - 2 = 13 – 2

│2x - 5│ = 11

2x – 5 = 11 (or) 2x – 5 = -11 

2x = 16 (or) 2x = -6

x = 8 (or) x = -3

Problem 11 :

│4x - 12│ + 8 = 0

Solution :

│4x - 12│ + 8 = 0

│4x - 12│ + 8 - 8 = 0 – 8

│4x - 12│ = -8

Absolute value cannot be less than 0. So there is no solution.

Problem 12 :

│7 - x│ + 2 = 12

Solution :

│7 - x│ + 2 = 12

│7 - x│ + 2 - 2 = 12 – 2

│7 - x│ = 10

7 - x = 10 (or) 7 - x = -10

-x = 3 (or) -x = -17

x = -3 (or) x = 17

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