91Ó°ÊÓ

An ordered pair, like \(0, 0\), is a set of values that denote specific points on a coordinate plane. In mathematics, these pairs are typically used to represent the values \(x, y\). Here, the first number in the pair represents the x-coordinate, and the second one represents the y-coordinate. Ordered pairs are fundamental when grappling with inequalities and equations because they allow us to test whether certain values satisfy a given condition.

For example, to check if an ordered pair is a solution to the inequality \(-x + 4y > 7\), we substitute the values of \x\ and \y\ from the ordered pair into the inequality. If the inequality holds true after substituting, the pair is a solution. In this sense, ordered pairs allow us to explore all possible values and identify which ones meet our criteria.

Substitution is a crucial process when solving or verifying inequalities with ordered pairs. It involves replacing the variable in the inequality with numbers from an ordered pair. This makes it possible to specifically evaluate whether those numbers satisfy the inequality's condition.

Let's break that down further: Take the inequality \(-x + 4y > 7\). Now, consider the ordered pair \(0, 0\). By substituting \x = 0\ and \y = 0\, we reframe the inequality into a numerical expression: \(-(0) + 4(0) > 7\).

Simplifying gives us \0 > 7\. This method is an efficient strategy in solving mathematical problems. It allows direct evaluation and avoids complex algebraic manipulation. Remember, substitution is not just a mechanical step; it's a powerful tool to verify solutions quickly and precisely.

After substituting the values from the ordered pair into the inequality, our next task is to evaluate if the resulting expression holds true. This checks if the pair indeed satisfies the condition set by the inequality. In our example, after substituting \(0, 0\) into \(-x + 4y > 7\), we obtained \0 > 7\.

Evaluation shows this statement is false because zero is not greater than seven. Therefore, the ordered pair \(0, 0\) is not a solution for the inequality.

The process of determining whether a pair is a solution involves simple logical reasoning to see if the inequality's truth value holds after substitution.

• If after substitution the inequality is true, the ordered pair is a solution.
• If the statement is false, the pair is not a solution.

This definitive check reinforces the role of inequalities in determining possible outcomes efficiently.