Absolute Value Equations

Absolute Value Equations Absolute Value Equations Before we start solving "Absolute Value Equations", we must understand actually what do we mean by "Absolute Value “. By "Absolute Value� , we mean the positive value of any number and it only represents the magnitude of any numerical value.

This simply means that the absolute value does not represent any direction and has a positive value every time. We use the symbol | x | to represent the absolute value of x, where x can be any integer.

When we put two parallel bars, to represent the absolute value of any integer, it simply indicates that we are talking about the positive value of that integer, without considering if the given variable has a positive value or a negative value. We say that the absolute value of any variable will be a positive number, every time.

So if we write | -4 |, the result is 4, also the value of | 4 | is also 4.

Again we must remember that - | 4| and the value of | -4 | are always different. Here the result of the first value will be a negative value i.e. -4 and the value of | - 4 | is always positive i. e. + 4

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If we take the "Absolute Value Equations", following the rules of solving the "Absolute Value “ we can solve the "Absolute Value Equations".

In geometrical terms absolute value of any number x is the distance x units from the origin without noticing the direction in which the distance is taken.

Let us now take the "Absolute Value Equations".

If we have the equation say | 2x – 3 | - 4 = 7 Now if we solve this equation, we get | 2x – 3 | = 7 + 4 Or we write | 2x – 3 | = 11 Now we say that as the left side of the equation has absolute value, it means that any of the following two equations are true : -

( 2x – 3 ) = 11

or (2x – 3 ) = 11

-

2x + 3 = 11

-

2x = 11 – 3 or 2x = 14

-

2x = 8 or

or 2x = 11 + 3

2x = 14

So we write X = -8/2

or x = 14 / 2

X = -4

or x = 7

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Thus we conclude that either the value of x will be -4 or we have the value of x = 7.

Now we can also check this value by plotting the two lines on the graph. Two lines will be plotted, in first case, we will take the value of x = -4 and the value of y will be taken as | 2x – 3|-4 Also we need to plot another equation, where we take the value of x = 7 and the value of y will be again taken as | 2x – 3 | - 4.

We observe that the two lines so formed on the graph represent the two different lines, from which one lies on the positive side and the another lies on the negative side of the number line.

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