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 | x² - 5x + 6 | = 8

Solution :
| x² - 4x - 5 | = 7

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

x² - 4x - 5 = 7    or     x² - 4x - 5 = - 7

x² - 4x - 5 – 7 = 0    or     x² - 4x - 5 + 7 = 0

x² - 4x -12 = 0    or     x² - 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 x² - 4x + 2 = 0



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

Using quadratic formula,

       4 ± √[4² - 4(1)(2)]          4 ± √ 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 ±√2}

So the solution of the absolute value equation is { 6,-2, 2 ±√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