Problem 45 - Algebra and Trigonometry

Question

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} x^{2}+y^{2} &<9 \\ x+y &>0 \\ x & \leq 0 \end{aligned}\right. $$

Step-by-Step Solution

Answer

Vertices are approximately \((0, 3)\) and \((0,0)\). The solution is bounded.

Step 1: Understanding Inequality Constraints

The first inequality is a circle, given by \( x^2 + y^2 < 9 \), meaning that the solution set is the interior of a circle with radius 3 centered at the origin \((0,0)\). The second inequality \( x + y > 0 \) represents the region above the line \( y = -x \). The third constraint \( x \leq 0 \) indicates that we consider only points where \( x \) is non-positive, or to the left of the \( y \)-axis.

Step 2: Graphing the Inequalities

To graph \( x^2 + y^2 < 9 \), draw a dotted circle with radius 3 centered at the origin. The region inside the circle is where this inequality holds. For \( x + y > 0 \), plot the line \( y = -x \) with a dotted line, and shade the region above this line. For \( x \leq 0 \), shade the entire left half of the coordinate plane (where \( x \) is negative or zero). The solution to the system of inequalities is the intersecting region of these three areas.

Step 3: Identifying the Intersection Regions

The intersection of all three inequalities will create a bounded region. Since \( x^2 + y^2 < 9 \) only includes points inside the circle and \( x \leq 0 \) constrains us to the left of the \( y \)-axis, the solution is a semi-circular region on the left side, above the line \( y = -x \) but inside the circle.

Step 4: Finding the Vertices

Since the region is curved (part of a circle), there are no literal vertices as in polygons. However, the boundaries are \( (0, 3) \) along the circle, the origin from the line and \( (0, 0) \) where the y-intercept of \( y = -x \) intersects the \( y \)-axis inside the circle.

Step 5: Determining if the Solution Set is Bounded

Since the region of the solution is part of a bounded circle, the solution set is bounded by the circle \( x^2 + y^2 = 9 \) on one side and the line \( x + y = 0 \) on another.

Key Concepts

Intersection of Inequalities

When dealing with a system of inequalities, we are searching for a region that satisfies all conditions at the same time. This common area is known as the intersection of the inequalities. In our exercise:

• The circle defined by the inequality \( x^2 + y^2 < 9 \) includes all the points inside the circle of radius 3 at the origin.
• The inequality \( x + y > 0 \) represents the region above the line \( y = -x \).
• Finally, \( x \leq 0 \) means we are only looking at the left side of the \( y \)-axis.

The intersection of these three inequalities describes a specific part of the coordinate plane, fulfilling all these conditions simultaneously. It is important to visualize each region step-by-step to identify where they overlap.

Graphing Inequalities

Graphing inequalities involves a few specific steps. Here's how you can graph the inequalities from the exercise:- **For \( x^2 + y^2 < 9 \):** Start by drawing a dotted circle with a radius of 3 units centered at the origin, which represents all the points where the sum of the squares of \( x \) and \( y \) is less than 9. The region inside this circle reflects the solutions to this inequality.- **For \( x + y > 0 \):** Plot the line \( y = -x \), also with a dotted line, because the inequality is strict (">"), not allowing for equality. Shade the region above this line, as it includes the points that solve this part of the inequality.- **For \( x \leq 0 \):** Shade the half-plane left of the \( y \)-axis because we only consider non-positive \( x \) values.The overlapping region from these steps represents the solution to the system of inequalities.

Circle Inequalities

A circle inequality, like \( x^2 + y^2 < 9 \), is an easy way to represent the inside of a circle mathematically. In this inequality:

• The expression \( x^2 + y^2 \) calculates the square of the distance from any point \( (x, y) \) to the origin \( (0,0) \).
• The "<" symbol indicates that we are interested in points strictly inside the circle.

Understanding circle inequalities helps in graphing them correctly. The circle in the inequality mentions only an upper bound, showing we stay inside the circle without reaching the boundary. The concept of inner versus outer regions plays a crucial role in finding the intersection when combined with other inequalities.

Bounded Solution Set

The term "bounded solution set" refers to a region that is enclosed within certain limits and does not extend infinitely in any direction. For the problem at hand, the bounded region is formed:

• By the circle \( x^2 + y^2 = 9 \) which caps the space on one side, providing a finite, circular boundary.
• By the line \( x + y = 0 \), acting as another definite boundary.

Since the solution is limited in all directions by the circle and line, it is finite and thus bounded. This is particularly clear because the solution region is within a semicircle on the left side of the \( y \)-axis, cut by the line \( y = -x \). Identifying whether a solution set is bounded or not helps to understand the nature of the whole system of inequalities, and subsequently, its solutions.