Problem 43 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution to the system of inequalities is the semi-circle region starting from (0,2) to (4,0) above the line \(x+y=2\), excluding the boundary line.

Step 1: Graph the Circle

Start by graphing the first inequality \(x^{2}+y^{2} \leq 16\). This represents a circle with radius 4 centered at the origin and covers all the points inside the circle including the boundary, as the inequality is less than or equal to.

Step 2: Graph the Line

Next, graph the linear inequality \(x+y>2\). It represents all the points above the line \(x+y=2\), not including points on the line, as the inequality is strictly greater than. The line passes through points (0,2) and (2,0).

Step 3: Identify Intersection

The solution to the system of inequalities is the region where both inequalities overlap. In this case, it is the semi-circle starting from (0,2) to (4,0) above the line \(x+y=2\), excluding the boundary line.

Key Concepts

Circle Inequalities

Circle inequalities often describe regions on the coordinate plane that are circular in shape. When we have an inequality like \(x^{2} + y^{2} \leq 16 \), it represents a circle with a specific radius centered at the origin. This inequality uses "less than or equal to" (≤), which means we include the points exactly on the circle as well as those inside it.

• For \(x^{2} + y^{2} \leq 16\), the circle's radius is 4, since \(\sqrt{16} \) is 4.
• A circle centered at the origin with radius 4 covers every point (x, y) such that when squared and summed, the value is less than or equal to 16.

When you graph this, use a solid line for the boundary because the inequality includes equal to (≤) and shade the entire interior of the circle. This shading represents all the possible solutions that satisfy the inequality.

Linear Inequalities

Linear inequalities represent a half of the coordinate plane divided by a straight line. With an inequality like \(x + y > 2 \), you are looking at all the points on one side of the line defined by the equation \(x + y = 2\). This line can be easily graphed by finding two points that satisfy the equation, such as (0,2) and (2,0).

• The equation \(x + y = 2\) forms the boundary, and points like (0,2) and (2,0) help draw this line accurately.
• Since the inequality is \(>\) (greater than), it does not include points on the line, hence we use a dashed line.

To determine which side to shade, pick a test point not on the line, such as the origin (0,0). If the inequality holds true for this test point when substituted, shade that side. For \(x + y > 2\), substituting (0,0) gives \(0 > 2\), which is false, so we shade the opposite side of the line.

Solution Set

A solution set for a system of inequalities refers to the region where all conditions are met simultaneously. The task is to find where the shaded region from both the circle inequality \(x^{2}+y^{2} \leq 16\) and the linear inequality \(x+y>2\) overlap.

• The circle's shading covers all points inside and on the circle centered at the origin with a radius of 4.
• The line cuts through the circle, and the shading for \(x + y > 2\) includes all points above this line.

The overlap happens in the upper part of the circle, above the line \(x+y=2\) but does not include this line due to the strict inequality (\(>\)). The visual result is a region that looks like half a "moon" shape continuing from the boundary at (0,2) to (4,0) and including all areas within the circle except the dividing line, as these points do not contribute to the solution set.