Question

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x<2 y-y^{2} \\ 0

Short answer

The solution to the system of inequalities is the region on the left side of the y-axis (x<0), below the curve of the parabola and above the line.

Step-by-step solution

Graph the First Inequality \(x

To graph this inequality, it might be easier to rewrite it in the form of \(x=y(2-y)\) as \(y=\sqrt{2-x}\) and \(y=-\sqrt{2-x}\). It's a downwards facing parabola once graphed. However, keep in mind that the inequality is 'less than', so all the x-values under the curve of the parabola are included to the solution set of this inequality.

Graph the Second Inequality \(0

To graph the second inequality, it would be easier if we rewrite it as \(y<-x\). It'll be a straight line passing through origin, with a negative slope. However, since the inequality is 'greater than', the solution set for this inequality will be the area above the line.

Find the Overlapping Region

Now that the graphs of both inequalities have been plotted on the Cartesian plane, the common solution would be the overlapping region or intersection of the two solution sets. You can clearly see the overlapping region on the left side of the y-axis (i.e. when x<0).

Key concepts

Graphing Inequalities

When it comes to understanding systems of inequalities, graphing is a powerful visual tool. It helps in observing not just individual solutions, but also the relationships between multiple inequalities. Each inequality in a system is graphed separately on a coordinate plane. The combined solution is found in the intersection of their solution sets.

• Start by transforming inequalities into equations to sketch their graphs.
• Use solid or dashed lines depending on whether the inequality sign includes equal to (≤ or ≥ for solid, < or > for dashed lines).
• For each inequality, shade the area representing solutions; it indicates which side of the line or curve satisfies the inequality.
• The overlapping shaded region of all inequalities represents the final solution set.

This visual approach not only enhances understanding but also fosters intuition about how inequalities interact and the nature of their solutions.

Parabola

A parabola is a U-shaped curve that is the graph of a quadratic function or equation. It has some distinctive characteristics:

• It can open upwards or downwards depending on the sign of the leading coefficient.
• The highest or lowest point of a parabola is known as the vertex.
• The axis of symmetry passes through the vertex and divides the parabola into two mirror images.
• A downward-opening parabola is typically represented by equations like \(y = ax^2 + bx + c \) where \(a < 0\).
• For the inequality \(x < 2y - y^2\), rewriting it as \(x = y(2-y)\) helps to visualize it as a down-facing parabola, indicating that solutions lie beneath the curve.

Understanding the shape and properties of a parabola makes it easier to approach graphing tasks, particularly when determining solution regions.

Linear Inequality

Linear inequalities involve expressions like ax + by < c, similar to linear equations but instead use inequality signs (<, >, ≤, ≥). Graphing linear inequalities involves a few steps that make it clear where solutions lie:

• Begin by graphing the related equation (e.g., y = mx + b) which serves as the boundary line.
• Convert the inequality into a form that makes graphing intuitive; for example, from \(0 < x+y\) to \(y < -x\), indicating the line crosses through the origin with a negative slope.
• An inequality like \(y < -x\) means shading below the line, as this area contains all the points where x and y satisfy the inequality.
• Since this line divides the plane, the solution region is either the half-plane above or below the boundary line.

Recognizing how linear inequalities shape solution sets enhances problem-solving abilities and makes it simple to interpret or construct graphs of systems of inequalities.