Problem 19 - Algebra and Trigonometry

Question

Systems of Equations and Inequalities. $$y

Step-by-Step Solution

Answer

The inequality \(y < x^{2} - 1\) represents a region below the parabola defined by \(y = x^{2} - 1\), excluding the parabola curve itself. The main steps of solution involve identifying the boundary curve and determining the solution set region through the symbol of inequality. The solution set can be graphically represented by shading the region below the parabola.

Step 1 Title: Standard Form of Equation

The equation of the parabolic curve that represents the border of the solution set is obtained by replacing the inequality sign with an equal sign in the inequality \(y < x^{2} - 1\). Thus, the equation of the border curve is \(y = x^{2} - 1\).

Step 2: Identify the Inequality Region

The original inequality is \(y < x^{2} - 1\), which means that all the points above the curve \(y = x^{2} - 1\) are not in the solution set and all the points below the curve are in the solution set. This is confirmed by the less than symbol in the inequality.

Step 3: Graph the Inequality

Plot the curve \(y = x^{2} - 1\) on a graph. This is a parabola that opens upwards with a vertex at the point (0, -1). All the points below the curve are shaded to represent the solution set. It is important to note that the curve itself is not part of the solution due to the strict inequality.

Key Concepts

Parabolic Curve

A parabolic curve is a U-shaped, symmetrical curve found in quadratic equations. In the context of systems of equations and inequalities, it often serves as a boundary for solution sets.
Imagine a smooth arch opening upwards or downwards. For these systems, this curve is defined by a quadratic equation of the form \(y = ax^2 + bx + c\).
In the exercise example, the equation \(y = x^2 - 1\) represents this boundary.

• The curve opens upwards because the coefficient of \(x^2\) is positive.
• The vertex, the lowest point on the parabola, is at (0, -1) since the equation is in the form \(y = x^2 - 1\).

This mathematical shape is fundamental for understanding inequality regions, as it marks the transition from one solution region to another.

Graphing Inequalities

Graphing inequalities involves sketching the boundary and shading the region that satisfies the inequality. After formulating the boundary equation from the inequality, such as \(y = x^2 - 1\) from \(y < x^2 - 1\), the graph of this equation needs to be plotted.
In graphing, a solid line indicates that points on the line are included, whereas a dashed line shows they are not part of the solution. Because the inequality is \(y < x^2 - 1\), we use a dashed line to denote that the points on the curve itself are not included in the solution set.

• Draw the parabolic curve \(y = x^2 - 1\) as the boundary.
• Shade the area below this curve, indicating the range of \(y\) values that make the inequality true.

Visualizing the shaded area helps in understanding which part of the graph represents valid solutions, ensuring clear identification of the region that satisfies the inequality.

Solution Set Identification

Identifying the solution set in systems of inequalities is about understanding which values satisfy all the given conditions. With the inequality \(y < x^2 - 1\), the solution set includes all points that fall below the parabola on a graph.
This inequality implies that any point with coordinates \( (x, y) \) where \( y \) is less than the corresponding \( x^2 - 1 \) value on the curve is part of the solution.
When looking at inequalities, it's vital to check:

• If the inequality is "less than" or "greater than," which indicates open boundaries, signified by dashed lines.
• If the inequality includes "equal to," suggesting closed boundaries, where the curve is solid.

In our example, since the inequality is a strict "less than," the curve itself isn't included, but every point below it is. Understanding and marking these distinctions are key to mapping out the right solution region.