anemone said:Find all real numbers \(\displaystyle a\) such that the equation \(\displaystyle |x-|x-|x-4||| =a
\) has exactly three real solutions.
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.
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.
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.
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.
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.
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.