In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x²+y²≤16, x+y>2

Verified step by step guidance

1. Step 1: Identify the inequalities in the system. The first inequality is $x^2+y^2\backslash leq\; 36$, which represents all points (x, y) inside or on the circle centered at the origin with radius 6, since \( \sqrt{36} = 6 \).
2. Step 2: The second inequality is $3x\; +\; y\; >\; 6$. This represents all points above the line $y\; =\; -3x\; +\; 6$. The line itself is not included because the inequality is strict (>).
3. Step 3: To graph the solution set, first draw the circle with center at (0,0) and radius 6. This circle includes all points where $x^2\; +\; y^2\; =\; 36$ and the interior where .
4. Step 4: Next, graph the line $y\; =\; -3x\; +\; 6$. Since the inequality is strict, use a dashed line to indicate points on the line are not included.
5. Step 5: Finally, shade the region inside the circle that lies above the line $y\; =\; -3x\; +\; 6$. The solution set is the intersection of these two regions, meaning points that satisfy both inequalities simultaneously.

Key Concepts

Graphing Circles and Inequalities

The inequality x² + y² ≤ 36 represents all points inside or on the circle centered at the origin with radius 6. Understanding how to graph this circle and shade the region that satisfies the inequality is essential for visualizing the solution set.

Graphing Linear Inequalities

The inequality 3x + y > 6 represents all points above the line 3x + y = 6. Knowing how to graph the boundary line and determine which side satisfies the inequality helps in identifying the feasible region for the system.

Solution Set of Systems of Inequalities

The solution to the system is the intersection of the regions satisfying each inequality. Understanding how to find and graph this overlapping region is crucial to determine where both conditions hold true simultaneously.