Question

Graph the solution set of each system of inequalities.3x + 5y ≤ 15x2 + y2 \u003c 9

Accepted Answer

Step 1: Identify the inequalities in the system. The first inequality is $$4y - 6x \leq 15$$ and the second inequality is $$x^{2}$$ + $$y^{2}$$ \u003c 16$$. Step 2: Rewrite the first inequality in slope-intercept form to better understand the boundary line. Solve for y$:

4y$$ \leq 6x + 15$$

y$$ \leq \frac{6}{4}x + \frac{15}{4}$$

which simplifies to

y$$ \leq \frac{3}{2}x + \frac{15}{4}$$. Step 3: Graph the boundary line y$$ = \frac{3}{2}x + \frac{15}{4}$$. Since the inequality is $$\leq$$, the boundary line is solid, indicating points on the line satisfy the inequality. Shade the region below this line because y$ is less than or equal to the expression. Step 4: For the second inequality x$$^{2}$$ + y$$^{2}$$ \u003c 16$$, recognize this as the interior of a circle centered at the origin $$(0,0)$$ with radius $$4 ($$since $$\sqrt{16} = 4). $$The inequality is strict ($$\u003c$$), so the boundary circle is dashed, and the solution includes all points inside the circle but not on the circle. Step 5: The solution set to the system is the intersection of the two regions: the area inside the circle $$x^{2}$$ + $$y^{2}$$ \u003c 16$$ and the area below or on the line y$$ \leq \frac{3}{2}x + \frac{15}{4}$$. Graph both and shade the overlapping region to represent the solution set.

Graph the solution set of each system of inequalities.
3x + 5y ≤ 15
x² + y² < 9

Verified video answer for a similar problem:

Key Concepts

Graphing Linear Inequalities

Graphing Circles and Circular Inequalities

Solution Set of a System of Inequalities

Graph the solution set of each system of inequalities.3x + 5y ≤ 15x2 + y2 < 9