Problem 47 - Algebra and Trigonometry

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} {x^{2}+y^{2} \leq 16} \\ {x+y>2} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the overlapping shaded region inside the circle and above the line.

Step 1: Graph the circle

The first inequality, \(x^{2} + y^{2} \leq 16\), represents a circle centered at the origin with a radius of 4 units. The inequality symbol 'less than or equal to' means that the area inside the circle is included in the solution. Thus, graph this circle and shade the region inside.

Step 2: Graph the line

The second inequality, \(x + y > 2\), is a line. We can rearrange it to the form \(y = -x + 2\), which has a y-intercept of 2 and a slope of -1. The 'greater than' sign indicates that the area above the line is included in the solution. Graph this line, do not include the line (so, plot it as a dashed line) and shade the region above.

Step 3: Identify the solution region

The solution of the system of inequalities is where the shaded regions of the circle and linear inequality overlap.

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing inequality expressions on a coordinate plane. These inequalities define specific regions on the plane that satisfy the given conditions. When drawing inequalities, it’s important to remember:

• Use a solid line for inequalities that include 'equal to' (like \(ackslash leq \) or \(ackslash geq \)). This indicates that points on the line satisfy the inequality.
• Use a dashed line for strict inequalities (like \( > \) or \( < \)). This shows that points on the line itself do not meet the criteria of the inequality.
• Shade the appropriate region that represents the solution set. For instance, above the line for an inequality like \(yackslash >...\).

In this exercise, you'll see how these concepts apply to both linear and circular inequalities, helping you visualize solutions in a more interactive way.

Solution Set

A solution set is essentially a collection of all points that satisfy the given system of inequalities. It's the intersection where shading from different inequalities overlaps on the graph. This overlapping region represents all possible solutions to the inequalities in question.

To find the solution set:

• First, graph each inequality individually, considering the type of line (dashed or solid) and which area to shade.
• Next, observe where the shaded regions from both graphs overlap. These overlapping areas are your solution set.

It's crucial to only consider points that satisfy all the inequalities in the system simultaneously. In this exercise, you'll explore how combining these areas reveals a solution that fits both the circle and line criteria.

Circle Inequality

A circle inequality often represents a circle on the coordinate plane, but with a region inside (or outside) included depending on the inequality symbol.
The inequality \(x^2 + y^2 \leq 16\) describes a circle centered at the origin (0,0) with a radius of 4, derived from \sqrt{16}\. This includes all points within and on the boundary of the circle since the inequality symbol is 'less than or equal to'.

When graphing:

• Draw a solid circle to include the points on the boundary.
• Shade the inside of the circle, indicating that all these points satisfy the inequality.

Understanding circle inequalities helps visualize circular regions, which is especially useful in geometry-related problems.

Linear Inequality

Linear inequalities define regions above or below specific lines on the graph. They take a typical linear equation format, but with inequalities instead of an equal sign.
For example, the inequality \(x + y > 2\) can be rewritten as \(y = -x + 2\) and graphed to show where the inequality holds true. The line itself represents the boundary, but:

• Since we're dealing with a 'greater than' symbol, plot a dashed line.
• Shade above this line to represent all points where the inequality is true.

Linear inequalities are fundamental in mathematics, providing a clear method to visualize and understand areas that satisfy certain conditions or constraints.