Absolute Value Equation

Absolute Value Equation : When you take the absolute value of a number, you always end up with a positive number (or zero). Whether the input was positive or negative (or zero), the output is always positive (or zero).

For example, | 3 | = 3, and | –3 | = 3 also.
This property — that both the positive and the negative become positive — makes solving absolute equation a little tricky. But once you learn the method, then its very easy.
Solved examples on absolute value equation :

1) |x| = 3

Then x may be positive or negative 3.

So x = 3 or x = -3
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2) Solve | x + 3| = 7

Solution :
To remove the absolute sign we have to make two equations i.e. two possible cases

x + 3 = 7 or
x + 3 = -7

x = 7 – 3 or
x = -7 – 3

x = 4 or x = -10

Then the solution is x = 4,-10

To confirm, check the following graph


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3) Solve |2x + 3| = 9

Solution :
|2x + 3 | = 9

To remove the absolute sign we have to make two equations i.e. two possible cases

2x + 3 = 9 or 2x + 3 = - 9

2x = 9 – 3 or 2x = - 9 – 3

2x = 6 or 2x = -12

x = 6/2 or x = -12/2

x = 3 or x = -6

Then the solution is x = 3,-6.

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4) Solve | 3x – 4 | - 5 = 18

Solution :
| 3x – 4 | - 5 = 18

| 3x – 4 | = 18 + 5

| 3x – 4| = 23

To remove the absolute sign we have to make two equations i.e. two possible cases

3x – 4 = 23 or 3x – 4 = - 23

3x = 23 + 4 or 3x = -23 + 4

3x = 27 or 3x = -19

x = 27/3 or x = -19/3

x = 9 or x = -19/3

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5) Solve | x
2 - 5x + 6 | = 8

Solution :
| x
2 - 4x - 5 | = 7

To remove the absolute sign we have to make two equations i.e. two possible cases

x
2 - 4x - 5 = 7 or x 2 - 4x - 5 = - 7

x
2 - 4x - 5 – 7 = 0 or x 2 - 4x - 5 + 7 = 0

x
2 - 4x -12 = 0 or x 2 - 4x + 2 = 0

Find the factors

(x – 6) ( x + 2) = 0

x – 6 = 0 or x + 2 = 0

x = 6 or x = -2

Now the 2nd equation x
2 - 4x + 2 = 0



a = 1 , b = -4 and c = 2

Using quadratic formula,

4 ~+mn~ √[4
2 - 4(1)(2)] 4 ~+mn~ √ 8
x = --------------------------= ---------
2(1) 2

4 + 2√2 4 - 2√2
x = ------ or x = ------
2 2

2(2 + √2) 2(2 - √2)
x = ------ or x = ------
2 2
Solution is { 2 ~+mn~√2}

So the solution of the absolute value equation is { 6,-2, 2 ~+mn~√2}

To confirm see the graph below



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Integers

Absolute value of Integers
Absolute Value Equation
Addition of Integers
Multiplication of Numbers
Division of Numbers
Number System

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