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Algebraic inequalities are fundamental components of mathematics that represent the relationship between expressions that are not equal. In essence, they determine the relative magnitude of two quantities. When solving inequalities, the goal is often to find the set of values that makes the inequality true.

It's important to be aware that operations affecting the entire inequality — such as multiplying or dividing by a negative number — can reverse the inequality sign. In the case of our exercise, we study a more complex inequality involving cubic and square terms. The provided solution illustrates how algebraic properties and geometrical concepts, like the triangle inequality, can be used in tandem to prove such inequalities.

In algebra, cubic terms are those raised to the third power, and square terms are raised to the second power. They are the building blocks of polynomial expressions and are especially significant in equations and inequalities.

Understanding the relationship between cubic and square terms often requires manipulating the given expressions to reveal underlying patterns or simplify the inequality. As shown in the solution, by setting up the right substitutions and rearrangements, the originally complex expression can be rewritten to show the link between these terms in a clear and manageable form.

The triangle inequality is a principle from geometry, but it also plays a critical role in algebraic contexts. It states that in any triangle, the length of a side is always less than the sum, and more than the difference, of the lengths of the other two sides. Mathematically, for any triangle with sides of lengths 'a,' 'b,' and 'c,' we have:
\[ a < b + c \] and \[|b - c| < a \] In the context of our inequality proof, the triangle inequality is used to assert that the sum of the squares of any two sides of a triangle is greater than the square of the third side. This helps us establish the inequality \[x^2+y^2+z^2 \geq 2(xy+yz+zx)\], paving the way for a mathematical simplification.

Mathematical simplification involves reducing complexity in mathematical expressions to make them easier to manage, interpret or solve. Simplification can involve factoring, combining like terms, or applying mathematical properties and theorems such as distributive, associative, and commutative laws.

In the context of our inequality, the substitution step that replaced 'a,' 'b,' and 'c' with 'x=a-b,' 'y=b-c,' and 'z=c-a' transformed the inequality into a more manageable form, enabling the use of the triangle inequality. This kind of simplification is crucial in mathematical proofs and solving problems because it makes the complex relationship between terms clear and often reveals the path to the solution.