Real Solutions for Equation |x-|x-|x-4||| = a

In summary, when asked to "find all real numbers a", one must find all possible values of the variable a that satisfy a given equation or inequality. This can be done through various methods such as algebraic manipulation, graphing, or using numerical methods. To solve for all real numbers a, one must have an equation or inequality with a variable a and use mathematical principles and techniques such as substitution, elimination, or factoring. Real numbers are numbers that can be represented on a number line, while complex numbers include both a real and imaginary component. Not all equations have real number solutions, as some involve complex numbers. To determine if all real number solutions have been found, one must follow appropriate steps and check that the solutions make sense in
  • #1
anemone
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Find all real numbers \(\displaystyle a\) such that the equation \(\displaystyle |x-|x-|x-4||| =a
\) has exactly three real solutions.
 
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  • #2
anemone said:
Find all real numbers \(\displaystyle a\) such that the equation \(\displaystyle |x-|x-|x-4||| =a
\) has exactly three real solutions.


The curve

\(\displaystyle y= |x-|x-|x-4||| \)

is piece-wise linear with knots at \(\displaystyle x=4/3,\ 2,\ 4\) and so considering the shape of the curve \(\displaystyle a=y(2)\) is the value sought.
.
 
  • #3
zzephod said:
The curve

\(\displaystyle y= |x-|x-|x-4||| \)

is piece-wise linear with knots at \(\displaystyle x=4/3,\ 2,\ 4\) and so considering the shape of the curve \(\displaystyle a=y(2)\) is the value sought.
.

The OP is having trouble logging on to MHB, and has asked me to congratulate you on her behalf on finding the correct answer, and says the method you have used is nearly identical to the method she used in finding the solution.
 

FAQ: Real Solutions for Equation |x-|x-|x-4||| = a

What does it mean to "find all real numbers a"?

When we say "find all real numbers a", we are referring to finding all possible values of the variable a that satisfy a given equation or inequality. This can be done through various methods such as algebraic manipulation, graphing, or using numerical methods.

How do I solve for all real numbers a?

To solve for all real numbers a, you need to have an equation or inequality with a variable a. Then, you can use mathematical principles and techniques such as substitution, elimination, or factoring to find the values of a that satisfy the given equation or inequality.

What is the difference between real numbers and complex numbers?

Real numbers are numbers that can be represented on a number line and include all rational and irrational numbers. Complex numbers, on the other hand, are numbers that include both a real and imaginary component and can be represented in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Can all equations have real number solutions?

No, not all equations have real number solutions. Some equations, known as imaginary or complex equations, have solutions that involve complex numbers. This means that the solutions cannot be represented on a number line and involve the imaginary unit i.

How do I know if I have found all real number solutions?

If you have followed the appropriate steps and methods to solve for all real numbers a, then you can be confident that you have found all possible solutions. It is important to check your work and make sure that your solutions make sense in the context of the original equation or problem.

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