The Absolute Value of a number is its distance from zero (on the number line). Thus, an absolute value is always positive. We indicate absolute value by putting two bars around the number.
NEGATIVE DISTANCE MAKES NO SENSE: You can be 50 meters away from your friend, but what would it mean if you were negative 50 meters away from your friend? That would be meaningless. Sure, you might be 50 meters to the other side of your friend, but you’re still 50 meters away. Think about absolute value the same way.
The most important thing to remember about absolute value is the concept of OR. Whenever you’re dealing with absolute value problems, remember that the stuff inside the absolute value brackets does not change
The most important thing to remember about absolute value is the concept of OR.
Whenever you’re dealing with absolute value problems, remember that the stuff inside the absolute value brackets does not change - the stuff on the other side of the equal sign does!!!
In other words: | X - 8 | = 4
Means that: X - 8 = 4 OR X - 8 = -4
Notice there’s no need to do anything to “X - 8?”
Do NOT touch anything inside the absolute value brackets! Never, ever, ever touch anything inside the absolute value brackets.
So long as you remember that there are always two solutions to any absolute value problem (there’s always an “or”) and that you change the non-bracketed side of the equation when converting from absolute value into two distinct equations, you’ll be all set!
Example: Solve for w, given that 12 + | w - 4 | = 30.
STEP 1: Isolate the expression within the absolute value brackets: 12 + | w - 4 | = 30 => | w - 4 | = 18
STEP 2: Once you have an equation of the form | x | = a & a > 0, you know that x = a. Remove the absolute value brackets and solve the equation for 2 different cases
STEP 3: Check to see whether each solution is valid by putting each one back into the original equation and verifying that the two sides of the equation are equal.
- In CASE 1, the solution, w = 22, is valid because 12 + | 22 - 4 | = 12 + 18 = 30.
- In CASE 2, the solution, w = -14, is valid because 12 + | -14 - 4 | = 12 + 18 = 30.
This is where most of our AP Guru students really seem to go bananas. Many students have a total freak out when they see something like this:
| X - 3 | > 6
To understand the concept, it is helpful to try to visualize the problem with a number line.
Absolute values can often more difficult to graph than the one above. Consider, for instance, the inequality | x + 2 | < 5. This seems more difficult, because the “ + 2 ” term seems to throw a wrench into our distance-on-the-number-line interpretation of this problem.
Notice that the center point for the possible values of X is now - 2, which is the value for x that fits X + 2 = 0. This is the “center point” for the number line graph. The distance from the center point (-2) to either end point remains the same.
From this example, we can extract a standard formula for interpreting absolute value. When | x + b | = c, the center point of our graph is -b The equation tells us that x must be exactly c units away from -b. Similarly, for the inequality | x + b | < c, the center point of the graph is -b , and the “less than” symbol tells us that x must be less than c units away from -b.
Example: What is the graph of | x - 4 | < 3?
The same is true for inequalities. However, remember the ONE key twist: the sign is reversed for the negative possibility!
In other words, to convert this into an “or” propositions: | X-3 | > 6
You get: X - 3 > 6 OR X - 3 < -6
Why do you reverse the sign when you do this? Rather than explain it verbally, let’s look at it mathematically. Let’s say you have this: | X | > 3
It means that “the distance of x from 0 on the number line is greater than 3.” That’s the precise definition of absolute value and of this inequality.
But what if you do this: X > 3 OR X > -3
That makes no real sense. So, x is greater than 3, but also greater than 3? The second piece of information doesn’t help at all...it’s useless. It’s like telling someone you’re taller than six feet but also taller than 5’9”.
But if we remember to switch the sign, we get: X > 3 OR X < -3
That makes way more sense! X is either greater than 3 or less than -3, meaning that its distance from 0 on the number line will remain larger than 3!
So long as you understand the concept of flipping the sign when given the negative “OR” possibility, you understand the hardest stuff you’ll ever be asked about absolute values on this exam.
To test your understanding, solve the following equation: 2 | X - 5 | + 3 > 19
2 | X - 5 | > 16
| X - 5 | > 8
Let’s try this one more time: Solve 3 | 2X + 4 | < 3X + 9
We get: | 2X + 4 | < X + 3
Next, I set up my “OR's”:
2X + 4 < X + 3 OR 2X + 4 > -X - 3
X + 4 < 3 OR 3X + 4 > -3
X < -1 OR X > -(7/3)
Therefore, X is less than -1 or greater than -(7/3). Done! X can be anything that isn’t between -1 and -(7/3).
So far, we have only looked at absolute value equations that have one unknown inside one absolute value expression. However, these equations can get more complicated by including more than one absolute value expression. The equation contains one variable and at least one constant in more than one absolute value expression.
Example: If | x - 2 | = | 2x - 3 |, what are the possible values for x?
However, note that case 1 and case 4 yield the same equation. Likewise, case 2 and case 3 yield the same equation. Thus, you only need to consider two real cases: one in which neither expression changes sign, and another in which one expression changes sign.
CASE A: Same Sign
(x - 2) = (2x - 3)
x = 1
CASE B: Different Signs
(x - 2) = - (2x - 3)
x = 5/3
You also have to check the validity of the solutions once you have solved the equations.
Both solutions are valid, because
|1 - 2 | = |2(1) - 3 | = 1, &
| 5/3 - 2 | = |2(5/3) - 3 | = 1/3
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