91Ó°ÊÓ

In the context of algebra, an ordered pair is a pair of numbers used to indicate the position of a point on a coordinate plane. The ordered pair is written in the form \( (x, y) \), where \( x \) is the horizontal value (also called the x-coordinate or abscissa), and \( y \) is the vertical value (y-coordinate or ordinate). These pairs are fundamental in graphing equations and inequalities.

When determining if an ordered pair is a solution for an inequality, like in our exercise \( 1.8 x - 3.8 y \geq 5 \), you essentially plug the values of the ordered pair into the inequality in place of \( x \) and \( y \) to check if the inequality holds true. If the result is true, then the ordered pair is a solution to the inequality.

The substitution method is a technique used to solve systems of equations, as well as to verify if an ordered pair is a solution for an equation or inequality. This method involves replacing variables with their corresponding values from an ordered pair or another equation.

In our example, the substitution method is applied by replacing \( x \) and \( y \) with the values given by the ordered pairs \( (0,0) \) and \( (1,-1) \). This process transforms the algebraic expression into a numerical statement that can be evaluated as true or false. For a thorough understanding, it's important to perform each substitution step carefully to avoid errors that might lead to incorrect conclusions about the solutions.

Determining inequality solutions involves finding all the ordered pairs that satisfy the inequality. An inequality like \( 1.8 x - 3.8 y \geq 5 \) specifies a set of points that forms a region on the coordinate plane. To verify individual solutions, as we did in our exercise, you use the substitution method to check if the ordered pairs make the inequality true.

For the ordered pair \( (0,0) \) the substitution didn't satisfy the inequality, making it not a part of the solution set. However, substituting \( (1,-1) \) resulted in a true statement, hence, it is included in the solution set. Understanding the concept of inequalities extends to recognizing regions, graphing, and analyzing the solutions visually as well, which are all critical skills in algebra.