91Ó°ÊÓ

When solving inequalities, the substitution method is a crucial technique that involves replacing variables with their given numerical values.

For example, consider the inequality involving variables, such as the one presented in the exercise:
\( y > 4x^2 - 7x \). When asked whether an ordered pair, in this case, \((2, -10)\), is a solution to the inequality, we apply the substitution method.

The process starts with replacing the variable \(x\) with 2 and \(y\) with -10. After substitution, we perform basic arithmetic operations to simplify the inequality. If the simplified inequality makes a true statement, the ordered pair is a solution; if false, it's not.

This step-by-step approach allows us to systematically determine the validity of the ordered pair concerning the inequality without plotting the graph of the function. It's like a quick 'reality check' that gives us immediate feedback on the validity of the given solution.

Ordered pairs are the foundation of coordinate systems and graphically represent solutions on a graph. These pairs \((x, y)\) delineate points on the Cartesian plane, where \(x\) is the horizontal component, and \(y\) is the vertical component.

In inequalities, we use ordered pairs to test if they are included in the set of solutions. Returning to our example \(y > 4x^2 - 7x\), the ordered pair \((2, -10)\) is at the heart of the exercise. You substitute the values into the inequality to verify whether they satisfy it.

Using ordered pairs to test solutions is a practical approach, especially when working with quadratic inequalities or plotting the graph. It gives a visual representation of where the pair lies in relation to the inequality's solution set.

Quadratic inequalities like the one in our exercise, \(y > 4x^2 - 7x\), involve a quadratic expression and an inequality sign (>, <, \(eq\), ≥, or ≤).

Solving these inequalities often involves finding the values of \(x\) that make the inequality true when plotted on a graph. The solutions form a region on the graph, either above or below the parabola (or between two parabolas), depending on the direction of the inequality.

When assessing a specific ordered pair, as we did in our example, we are essentially checking if that point falls within the solution region. Unlike linear inequalities, which form straight lines, the solution sets for quadratic inequalities can be more complex, often resembling a 'smile' or a 'frown' on the graph.

The ordered pair (2, -10) does not satisfy the quadratic inequality from the exercise, showing us that not all pairs will be part of the solution set.