Problem 63 - A Graphical Approach to College Algebra

Question

Graph the solution set of each system of inequalities by hand. Concept Check \(\quad\) Which one of the choices that follow is a description of the solution set of the following system? $$ x^{2}+y^{2} < 36 $$ \(y < x\) A. All points outside the circle \(x^{2}+y^{2}=36\) and above the line \(y=x\) B. All points outside the circle \(x^{2}+y^{2}=36\) and below the line \(y=x\) C. All points inside the circle \(x^{2}+y^{2}=36\) and above the line \(y=x\) D. All points inside the circle \(x^{2}+y^{2}=36\) and below the line \(y=x\)

Step-by-Step Solution

Answer

Answer: D. All points inside the circle and below the line.

Step 1: Understand the Circle Inequality

The inequality \(x^2 + y^2 < 36\) represents the set of all points inside the circle with center at the origin \((0, 0)\) and radius 6. This includes all points such that the sum of the squares of their \(x\) and \(y\) coordinates is less than 36.

Step 2: Understand the Line Inequality

The inequality \(y < x\) represents the region below the line \(y = x\). This means any point \((x, y)\) where the y-value is less than the x-value falls below this diagonal line.

Step 3: Determine the Overlapping Region

To find the solution set for the system of inequalities, determine the region where both conditions are satisfied. This means identifying points that are both inside the circle \(x^2 + y^2 < 36\) and below the line \(y = x\).

Step 4: Identify the Correct Description

From the analysis above, we need points that are inside the circle and below the line \(y = x\). The correct description matching this region is D: All points inside the circle \(x^2 + y^2=36\) and below the line \(y = x\).

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing regions on a coordinate plane that satisfy given conditions. By graphing, you can visually understand where certain equations hold true. For each inequality, you'll identify and shade the right area.

When graphing:

• Start by plotting the equation as if it were an equality (e.g., for \(y < x\), plot \(y = x\).
• Choose a test point not on the line to see which side satisfies the inequality.
• Shade the region that includes all points satisfying the inequality.

This visualization helps to see where two or more conditions overlap, finding the solution to a system of inequalities.

Circle Inequality

The circle inequality \(x^2 + y^2 < 36\) describes a set of points within a circle centered at \( (0, 0) \) with a radius of 6. This means all the points whose distance from the origin is less than 6 will satisfy this inequality.

Understanding this concept involves knowing:

• Standard circle equation: \(x^2 + y^2 = r^2\).
• In \(x^2 + y^2 < 36\), \(r = 6\), and you're looking for points that lie inside.

This inequality draws an open circle, which means it doesn't include the boundary circle itself.

Line Inequality

The inequality \(y < x\) represents all points that lie below the diagonal line \(y = x\). This line acts as a boundary separating where points are either greater or less than each other.

Key aspects to remember:

• The line \(y = x\) has a slope of 1, passing through the origin.
• For \(y < x\), you’re interested in the region where the \(y\)-value is less than the \(x\)-value.

Shading the area below this line on the graph visually shows all solutions to this inequality.