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Ordered pairs are a fundamental concept in coordinate geometry. They consist of two elements, often written in parentheses like \((-1, 4)\). Here, the first element represents the \(x\)-coordinate, and the second element represents the \(y\)-coordinate.

These pairs are essential for indicating specific points on a two-dimensional plane. For any given point \((x, y)\), the \(x\) value tells us how far to move horizontally from the origin, while the \(y\) value tells us how far to move vertically. This system helps in not only plotting points on a graph but also in solving mathematical problems like inequalities, as it provides a clear position for comparison.

Solution verification is a vital step in confirming that an ordered pair satisfies a given condition, such as an inequality. In our example, we must determine if \((-1, 4)\) is a solution to the inequality \(y < x^2 + 6x + 12\).

To do this, we substitute \(x = -1\) and \(y = 4\) into the inequality and then evaluate the quadratic expression. This computation involves several steps like squaring numbers and performing multiplication/addition:

• Calculate \((-1)^2\), resulting in \(1\).
• Multiply \(6\) by \(-1\), resulting in \(-6\).
• Sum the results: \(1 + (-6) + 12 = 7\).

After evaluating, we compare 4 with the result, 7. Since \(4 < 7\), the pair \((-1, 4)\) proves to be a true solution for the inequality. By repeating this process with different values, we can verify solutions efficiently across various scenarios.

Quadratic expressions form a key component of many mathematical concepts, including inequalities. A typical quadratic expression includes terms with \(x^2, x\), and a constant, as seen in \(x^2 + 6x + 12\).

Quadratic expressions like these behave differently than linear expressions. They can open upwards or downwards on a graph, forming parabolas, which introduce interesting challenges when dealing with inequalities.

In our context, the expression \(x^2 + 6x + 12\) must be evaluated at a specific \(x\) point:

• Substitute \(x = -1\) into the expression.
• Calculate \((-1)^2 = 1\).
• Add \(6(-1) = -6\).
• Add the constant \(12\), leading to \(1 - 6 + 12 = 7\).

The result is utilized in the inequality to check if the inequality holds true. Understanding how to manipulate and solve quadratic expressions is critical for mastering a broad array of mathematical challenges, including verifying solutions for inequalities.