91Ó°ÊÓ

Understanding ordered pair solutions is a fundamental aspect of algebra, particularly when dealing with systems of inequalities. An ordered pair, typically written as \( (x, y) \), represents a potential solution to a system of equations or inequalities. In the context of systems of linear inequalities, an ordered pair is a solution if, when substituted into each inequality, it satisfies all of them simultaneously.

For example, when evaluating the solution pair \( (6, 0) \) for the given inequalities, we substitute \( x=6 \) and \( y=0 \) into each inequality: \( 6^2 + 0^2 \), \( -3(6) + 0 \) and \( \frac{2}{3} (6) - 0 \). The ordered pair must make all inequalities true to be considered a solution. In our exercise, each tested ordered pair failed to satisfy at least one of the inequalities, thus they were not valid solutions.

Inequality graphing helps to visualize and solve systems of inequalities. Each inequality is represented by shaded regions on a coordinate plane, and the solution to the system is where these regions overlap. When graphing, we use a solid line if the inequality includes an equal to \( (\geq, \leq) \) part and a dashed line if it does not \( (>, <) \) to show the boundary.

Consider \( -3x + y \leq 10 \). To graph this inequality, we would first draw the line \( -3x + y = 10 \) as a solid line because of the \leq symbol. Then, we'd shade below this line as the inequality symbol \leq indicates that the region below the line is included. Graphing each inequality of our system would thus provide a visual representation of where the valid solution(s) may reside. It's a quick check to validate the correctness of ordered pair solutions.

Quadratic inequalities involve expressions of second-degree polynomials and are of the form \( ax^2 + bx + c > 0 \) or similar, with \leq, \geq, < symbols. The inequality \( x^2 + y^2 \geq 36 \) from the exercise is a quadratic inequality in two variables, representing all the points at or outside a circle with radius 6.

To graph such an inequality, we identify the conic section it represents (in this case, a circle), and shade the appropriate area. The inequality symbol \geq tells us to shade the area on or outside the circle's boundary. Solving quadratic inequalities analytically or by graphing will reveal the set of points that satisfy the inequality. In the context of our exercise, this demonstrates why certain ordered pairs did not qualify as they didn't fall within the satisfied region.

Linear inequalities, such as \( -3x + y \leq 10 \) and \( \frac{2}{3}x - y \geq 5 \) from the exercise, have properties similar to linear equations but with additional considerations due to the inequality symbols. For instance, when we multiply or divide an inequality by a negative number, the inequality sign reverses. Also, adding or subtracting the same number from both sides of an inequality preserves the inequality.

The properties are useful when solving or manipulating inequalities to find solutions. Understanding these properties can shed light on why certain steps, like checking each ordered pair's satisfaction of the inequalities, are carried out in the solution process. Additionally, these properties guide us in the systematic graphing of inequalities to identify solution regions on a coordinate plane.