Solving Equations with Moduli - Guidelines & Tips

  • Thread starter Astudious
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In summary: Substitute this value back into the original equation and solve for ##f(x)## and ##g(x)## using the method outlined before.Hope this helps.
  • #1
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How does one go about solving equations in one variable which contain moduli? For instance, those of the form f(x,|x|)=0 or f(x,|g(x)|) more generally.

Obviously I don't expect a completely "one-size fits all" solution, but a general approach to dealing with the moduli is what I'm looking for. (i.e. let's assume that, once the moduli are gone, I will be able to deal satisfactorily with the remaining equation.)
 
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  • #2
Astudious said:
How does one go about solving equations in one variable which contain moduli? For instance, those of the form f(x,|x|)=0 or f(x,|g(x)|) more generally.

Obviously I don't expect a completely "one-size fits all" solution, but a general approach to dealing with the moduli is what I'm looking for. (i.e. let's assume that, once the moduli are gone, I will be able to deal satisfactorily with the remaining equation.)

Get the modulus on one side, and everything else on the other side.

##|f(x)| = g(x)##

Using the definition of the absolute value, ##|a| = a## if ##a## is positive (or 0), and ##|a| = -a## if ##a## is negative, we have two equations:

##g(x) = f(x)##
##g(x) = -f(x)##

You might get extraneous solutions though, so always plug in the values you found into ##g(x)## and make sure ##g(x)## is positive (or zero). If you find that ##g(x)## is negative for a particular value, ignore this solution, since the absolute value of any real number is greater than or equal to zero by definition.

Hope this helps.

By the way, the functions ##f## and ##g## I used in my explanation are in no way related to those in your post, so don't get confused.
 
  • #3
MohammedRady97 said:
Get the modulus on one side, and everything else on the other side.

##|f(x)| = g(x)##

Using the definition of the absolute value, ##|a| = a## if ##a## is positive (or 0), and ##|a| = -a## if ##a## is negative, we have two equations:

##g(x) = f(x)##
##g(x) = -f(x)##

You might get extraneous solutions though, so always plug in the values you found into ##g(x)## and make sure ##g(x)## is positive (or zero). If you find that ##g(x)## is negative for a particular value, ignore this solution, since the absolute value of any real number is greater than or equal to zero by definition.

Hope this helps.

By the way, the functions ##f## and ##g## I used in my explanation are in no way related to those in your post, so don't get confused.

Thanks. So we just rearrange into the form above, and then solve the two equations

##g(x) = f(x)##
##g(x) = -f(x)##

and use the superset of the solutions, removing any solutions which lead to g(x)<0 since no modulus of f(x) can equal them.

What if it were something like

##|f(x)| + |g(x)| = h(x)##

?
 
  • #4
Astudious said:
Thanks. So we just rearrange into the form above, and then solve the two equations

##g(x) = f(x)##
##g(x) = -f(x)##

and use the superset of the solutions, removing any solutions which lead to g(x)<0 since no modulus of f(x) can equal them.

What if it were something like

##|f(x)| + |g(x)| = h(x)##

?
Astudious said:
Thanks. So we just rearrange into the form above, and then solve the two equations

##g(x) = f(x)##
##g(x) = -f(x)##

and use the superset of the solutions, removing any solutions which lead to g(x)<0 since no modulus of f(x) can equal them.

What if it were something like

##|f(x)| + |g(x)| = h(x)##

?

Square both sides.

##[|f(x)| + |g(x)|]^2 = h(x)^2##
##|f(x)|^2 + 2|f(x)||g(x)| + |g(x)|^2 = h(x)^2##

Recall that ##a^2 = |a|^2## and ##|a||b| = |ab|##

##f(x)^2 + 2|f(x)g(x)| + g(x)^2 = h(x)^2##
##|f(x)g(x)| = \frac{1}{2} [h(x)^2 - f(x)^2 - g(x)^2]##

You now have the equation in the form you mentioned in your first post.
 
  • #5


Solving equations with moduli can be tricky, but there are some general guidelines and tips that can help you approach these types of equations.

1. Understand the definition of modulus: The modulus of a number is its distance from zero on a number line. This means that the modulus of a positive number is the same number, while the modulus of a negative number is the number multiplied by -1. For example, the modulus of 5 is 5 and the modulus of -5 is also 5.

2. Identify the possible values of the variable: When dealing with moduli in equations, it's important to consider both positive and negative values for the variable. For example, in the equation f(x,|x|)=0, the possible values of x could be both positive and negative.

3. Use the definition of modulus to rewrite the equation: In some cases, you may be able to use the definition of modulus to rewrite the equation in a simpler form. For example, in the equation f(x,|x|)=0, you can rewrite |x| as x or -x, depending on the value of x. This will give you two separate equations to solve.

4. Consider the cases separately: In equations with moduli, it's often helpful to consider the two cases separately. For example, in the equation f(x,|x|)=0, you can consider the case where |x| is equal to x and the case where |x| is equal to -x. This will give you two separate equations to solve.

5. Use algebraic techniques to solve the equations: Once you have rewritten the equations without the moduli, you can use standard algebraic techniques to solve them. This may involve simplifying the equations, factoring, or using the quadratic formula.

6. Check your solutions: When solving equations with moduli, it's important to check your solutions to make sure they satisfy the original equation. Remember that for any value of x, the modulus of x is always positive. So if your solution for x is negative, you may need to change the sign to make it positive.

In conclusion, solving equations with moduli requires a combination of understanding the definition of modulus, rewriting the equations, and using algebraic techniques. It's also important to consider both positive and negative values for the variable and to check your solutions to ensure they satisfy the original equation. With these guidelines and tips, you should be able to approach equations
 

1. What is a modulus in terms of solving equations?

A modulus, denoted by the symbol |x|, is the absolute value of a number. It represents the distance of a number from zero on a number line.

2. How do you solve equations with moduli?

To solve equations with moduli, you need to isolate the modulus on one side of the equation and then consider two cases: when the value inside the modulus is positive and when it is negative. You can then solve for the variable in each case and combine the solutions.

3. Are there any specific guidelines to follow when solving equations with moduli?

Yes, there are a few guidelines to keep in mind when solving equations with moduli. First, always isolate the modulus and consider the two cases mentioned above. Second, when solving for the variable, you should use the positive value of the modulus if the value inside is positive, and the negative value if it is negative. Finally, always check your solutions by plugging them back into the original equation.

4. Can you provide an example of solving an equation with moduli?

Sure, let's solve the equation |2x-3| = 5. First, isolate the modulus to get two equations: 2x-3 = 5 and 2x-3 = -5. Solving each equation gives us x = 4 and x = -1. So the solutions are x = 4 and x = -1.

5. Are there any tips for solving equations with moduli efficiently?

One tip is to always check for extraneous solutions, which can occur when you square both sides of an equation with moduli. Another tip is to avoid simplifying the equation too early, as it can lead to errors. It is also helpful to practice identifying and solving equations with moduli to become more comfortable with the process.

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