Absolute Values

Absolute values

One of the most profound things I learned in my recent algebraic exorcism is that absolute values represent distances. More specifically, the distance of a given value to zero ( 0 ) in the number line. Since distances are always positive, working with absolute values would have you work with positive numbers. So, given any number:

| x |  =  x

x  =  x

which demonstrates that absolute values are always positive. Take note, however, that there are not negative absolute values, so such an statement is undefined:

| x |  =  - x

  Absolute values versus absolute value equations

Absolute value used and/or presented as equations are of a similar vein. Since one would be working with equations containing distances to the number zero in the number line, the values of such equations would yield two possibilities: one positive and one negative.

For example, consider the equation

4 | x |  -  4  =  -2 | x |  +  8

Here we need to solve for x, and we are given two absolute values to start with and some constants. The first step is to combine the constants and those with coefficients, so we multiply both sites by 2 |x|. This gets rid of -2 on the right hand side. Note that the negative sign applies to the coefficient 2 and not on the absolute value |x|.

2 | x |  +  4 | x |  -  4  =  - 2 | x |  +  8  +  2 | x |

6 | x |  -  4  =  8

Adding 4 to both sides would get rid of the constant on the left side

4  +  6 | x |  -  4  =  8 + 4

6 | x |  =  12

Finally, we multiply both sides to 1/6 (which is basically dividing both sides by 6) to arrive to the value for |x|

( 1/6 )  *  6 | x |  =  12  *  ( 1/6 )

| x |  =  2

And so we now have our answer, which can be interpreted as the distance of the value for this equation to zero (0) is two (2). We now have the distance, but we don't have the value yet. Not to say that getting the actual values are any real work, but it's worth noting the difference here. So we ask ourselves, what values in the number line, if any, have a distance of two (2) from the number zero (0)? Turns out, we have two possible values, namely:

x = 2 or x = - 2

  Absolute value expressions

In the previous case we have | x | as the absolute value, but in some cases we can have whole expressions, such as:

5 | x + 3 | - 3 = 7

So here we have | x + 3 | to denote the distance to zero ( 0 ), and that distance is being used by the left hand side to satisfy the equation somehow. As with the first example, this also yields two possible values, as we shall see in a moment. For now, let us first determine the value of | x + 3 | in the context of this equation:

5 | x + 3 | - 3 = 7

3 + 5 | x + 3 | - 3 = 7 + 3

5 | x + 3 | = 10

(1/5)  *  5 | x + 3 |  =  10  *  (1/5)

| x + 3 | = 2

Now we have the value for the absolute value expression, in this case a two ( 2 ). This is essentially saying that the expression is two steps away from zero; but, since it is an expression, we would need to simplify it further to get workable values that we can apply later on to check if we can satisfy the original equation.

So first we must realize that there are two ( 2 ) possible values that are actually two ( 2 ) steps away from zero ( 2 ), namely 2 and - 2.

| x |  =  2

x = 2  or  x = -2

Given that knowledge we can actually solve for the resuling absolute value expression in two ways. First is through the positive value:

x + 3  =  2

- 3 + x + 3  =  2 - 3

x  =  - 1

And then the negative:

x + 3  =  - 2

- 3 + x + 3  =  - 2 - 3

x  =  - 5

So now we have two possible values for x in the absolute value expression. Note that these possible values are for x, not for the expression itself. We already know the possible values for our absolute value expression, which is 2 and - 2.

The next step is to check these values agains the original expression to see if we actually get an equation. First, the positive value ( 2 ):

5 | - 1 + 3 |  -  3  =  7

5 | 2 |  -  3  =  7

(5 * 2)  -  3  =  7

10 - 3  =  7

7  =  7

And the negative value ( - 2 )

5 | - 5 + 3 |  -  3  =  7

5 | - 2 |  -  3  =  7

(5 * 2 )  -  3  =  7

10 - 3  =  7

7  =  7

  The zero distance

A small gotcha involves equations are equal to zero. Such equations are literally saying that the distance is zero; we just need to solve for the unknown to make it happen. We should note that the equation:

|x|  =  0

has one and only one value, and that is zero.

x  =  0

This is compared to the normal equations which has two possible values, a positive and a negative. So, given an equation like:

| 3x - 9 |  =  0

would have us solving for the number that will satisfy the right hand side with only one unique value. Also note that the absolute value here is the whole left side, and not just x. This conveniently fits into our definition for a absolute value being equal to zero.

3x  -  9  =  0

9  +  3x  -  9  =  0  +  9

3x  =  9

( 1/3 )  *  3x  =  9  *  ( 1/3 )

x  =  3

So this gives us a value that would make the whole left side equal to zero (0). Now we can check easily if this really satisfies our equation:

3 (3) - 9  =  0

9 - 9  =  0

0 = 0

To say "solve an equation", it usually is saying to find the value of x. Solve: |x - 1| = 5 First off, we isolate the absolute value expression. In this one, it is already done for us. Next, we create two cases: Case I and Case II. In Case I, we set the side that is not the absolute value expression to positive, and in Case II, we set the side that is not the absolute value expression to negative. With this done, we can take off the bars and solve for x. Case I: |x - 1| = 5 x - 1 = 5 x = 6 Case II: |x - 1| = -5 x - 1 = -5 x = -4 Our last step is to plug in both values into the original equation and find which one might be extraneous. Solve: |x - 3| = -4 When we isolate the absolute value expression, in this case it is done for us again, and the side that is not the absolute value expression is negative, we can automatically assume that this equation has no solutions. Because absolute values have to be positive. Solve: |x² - 2x| = 1 If the absolute value expression is quadratic, in both cases we set everything to one side and either factor or use the quadratic formula to solve for x. Case I: |x² - 2x| = 1 x² - 2x = 1 x² - 2x - 1 = 0 x = ~2.41 and ~0.41 Case II: |x² - 2x| = -1 x² - 2x = -1 x² - 2x + 1 = 0 x = 1 Remember to check and see if any are extraneous. Solve: |x - 10| + 10x = x² With this problem, it does not isolate the absolute value expression for us. So let us do this first. |x - 10| = x² - 10x Since there are two terms on the right side, we bracket them and consider one big term for each case. Then, since it is quadratic, we set each case to one side and solve for x. Case I: |x - 10| = x² - 10x x - 10 = x² - 10x 0 = x² - 11x + 10 x² - 11x + 10 = 0 x = 10 and 1 Case II: |x - 10| = -(x² - 10x) |x - 10| = -x² +10x x - 10 = -x² +10x x² - 9x - 10 = 0 x = 10 and -1 Remember to check and see if any are extraneous.