Absolute Value Equations
Absolute value equationsare equations involving expressions with theabsolute value functions. This wiki intends to demonstrate and discuss problem solving techniques that let us solve such equations.
A very basic example would be as follows:
Find all values of satisfying
Usually, the basic approach is to analyze the behavior of the function before and after the point where they reach 0. For example, for one could analyze the cases where or , or even if required. However, these problems are often simplified with a more sophisticated approach like being able to eliminate some of the cases, or graphing the functions. In this wiki, we intend to discuss this techniques along with strategies on when to use which.
Contents
Methodology
Introduction to absolute value equations
Methodology to solve absolute value equations: the techniques used to solve absolute value equations and when to use which one
Take an example to describe the following methodology:
1) Understanding absolute value--positive, negative case (or graph approach)
2) Determining possible solutions
3) Verifying solutions
Technique - Squaring Both Sides
Explain - How do we use this technique to solve absolute value equations?
Remember to verify the possible solutions - why and how?
2-3 examples in increasing order of difficulty - explaining how we squared both sides to solve more difficult problems
Followed by 1-2 TIY problems - relevant to be solved by case work technique
年代uppose we have an equation of the form . Since both sides are positive, we can square them without adding extraneous solutions: Then solve it as an ordinary equation: 年代o we see that or .
年代olve the equation for real .
We square both sides to obtain Here we don't need to expand both sides; just apply the difference of two squares to find the factors: The solutions are
Technique - Casework
Because absolute value can be defined as piecewise functions, depending on where the value of is with respect to the number line, you have to work with a different "piece" of the piecewise function.
General steps:
Using the definition of absolute value as a piecewise function, "undo" the absolute value sign(s) and write cases. For example, we know that the expression in the absolute value sign can either be positive or negative.
年代olve each case for .
Verify the solutions.
Find all real values of such that
We first isolate the absolute value onto one side:
Now, we "undo" the absolute value signs and split the equation into its two cases, the positive case and the negative case:
Find all real values of such that .
There are four possible cases, but one will be eliminated due to impossibility:
Case 1.If and are both positive, then
Case 2.If is negative and is positive, then However, when , and are both positive, so this is not a valid solution to the equation.
Case 3.If and are both negative, then
Case 4.If is positive and is negative, it is an impossible case. Graph the two lines if you are not convinced.
Therefore, the solution set is
Find all real values of such that
In this problem we are dealing with 3 terms of absolute values. Their turning points (the values of such that they change sign) of the three terms are respectively. Hence, we need to check the cases , , , .
Case 1.
In this case, the three terms will always be negative. Hence, However, is not within the domain . Thus this solution is not valid.Case 2.
In this case, the three terms will be negative, positive, and negative, respectively. Hence, lies between and . Thus is one of the solutions.Case 3.
In this case, the three terms will be positive, positive, and negative, respectively. However, Thus there is no solution within this domain.Case 4.
In this case, the three terms are always positive. Hence, which lies between and . Thus is another solution.In conclusion, and are the solutions for the given equation.
Find all real values of such that .
Case 1. both positive
Reject because it does not make both and positive.Case 2. negative, positive
We only asked for real solutions, so at this point we ignore this case because we're going to get imaginary results.Case 3. positive, negative
这是一个不可能的情况下(图线和哟u'll see why), so we can ignore it.Case 4. both negative
Because they're both negative, the negatives end up "canceling" and become positive, which was what we had in Case 1. However, the restriction is different from Case 1 (here, both and have to be negative, not positive ), so instead of rejecting ,我们reject from this case. Basically, in this specific case 4, is not a possible solution, but it does not mean it's not a possible solution for Case 1 because we're simply going piece by piece in this piecewise function--in the end we will take the union of all possible solutions.Thus, the solutions are .
Technique - Sketching Graph
年代ometimes absolute value equations have a ridiculous number of cases and it would take too long to go through every single case. Therefore, we can instead graph the absolute value equations using the definition of absolute value as a piecewise function. To get each piece, you must figure out the domain of each piece. This method is highly beneficial when the question writer asks for the number of solutions instead of the actual solutions. Let's work through some examples to see how this is done.
Find all real solutions to .
To graph this, there are two possible cases: when is positive, and when is negative.
When is positive? (Also, when , will be negative.)
We know that there will be a "turning point" at for the graph of .
Finally, using the definition of absolute value, we know that when , , and when , . We now just need to graph and look for the intersections.
You can see that the solutions are .
Another benefit of this graphing technique is that you do not need to verify any of the solutions--since we are only graphing the pieces that are actually mathematically possible, we get all the solutions we are looking for, no less and no more. If you could not discern the solutions from the picture, you can simply solve the equation for each case.
Find all real solutions to .
The possible cases are that
1. are both positive;
2. is negative and is positive;
3. are both negative.
We need to figure out the domains for which each of these holds.Case 1holds when .
Case 2holds when .
Case 3holds when .Now, let's write our piecewise function.
When ,我们have .
When ,我们have .
When ,我们have .
As you can see in the graph, the solutions for the given equation are .
Find all real solutions to .
To graph this, we again only look at the possible cases and when they would occur:
1. both positive
2. negative, positive
3. both negative.Case 1is true when .
Case 2is true when .
Case 3is true when .When and when ,我们have .
When ,我们have .
It is evident that the solutions are
Problem Solving - Miscellaneous
Any other technique (fact, definition) you can use to solve the problems? Otherwise move on to the followings.
3-4 examples solved by using a mix of more than one of above techniques
Add guiding text in between. Guiding text means phrasing the section in a way that it keeps on telling the reader what's going on in this section.
3-4 TIY problems - using multiple techniques to solve
What is the sum of all real numbers satisfying
Observe that Then the given equation becomes
If then we rewrite the equation to obtain
If then we rewrite the equation to obtain which do not satisfy the assumption Thus there are no solutions in this interval.
If then we rewrite the equation to obtain
Therefore, the above three cases give two solutions and the sum of which is
国际海事组织1959/2解方程 for real (where the square roots are only defined for non-negative values), when
- ;
- ;
- .
Here we don't see any absolute value involved with the equation. Before doing anything, note that our first restriction for is and for is . Intuitively, we could square both sides to get rid of some square roots: Great! We find a perfect square inside the square root, so an absolute value will appear: Now we are going to find the possible cases for :
When ,我们have Then, by our assumption of ,我们get that this solution only works when .
When , something interesting happens:
年代o, when , the equation becomes independent of , implying that any value of the interval will be a solution for the first point.
When , there are no solutions by our restriction of .
Finally, when we have
What happens if we allow the square roots to admit negative values?
年代ometimes, in minimization problems, it often helps us to see that the value of an expression inside the absolute value is at least 0.