Question

Graph each system of inequalities. $$\left\\{\begin{array}{l}x^{2}+y^{2} \leq 9 \\\x+y \geq 3\end{array}\right.$$

Short answer

The intersection region inside the circle centered at the origin with a radius of 3, above and including the line \(x + y = 3\).

Step-by-step solution

Graph the Circle Inequality

Graph the inequality \(x^2 + y^2 \leq 9\). This inequality describes a circle centered at the origin with a radius of 3. Since the inequality is \leq 9\, you will shade the inside of the circle to show all the points that satisfy this condition.

Graph the Linear Inequality

Next, graph the inequality \(x + y \geq 3\). Start by graphing the line \(x + y = 3\). This is a straight line with a slope of -1 and y-intercept of 3. Because the inequality is \geq 3\, you need to shade above the line to include all points that satisfy this condition.

Identify the Intersection Region

The solution to the system of inequalities is the region where the shaded areas from both individual inequalities overlap. This region is where both conditions \(x^2 + y^2 \leq 9\) and \(x + y \geq 3\) are true at the same time.

Key concepts

Graphing Inequalities

When graphing inequalities, the main goal is to find and shade the region on a coordinate plane that satisfies the given conditions. For each inequality, you'll draw its boundary line or curve first. If the inequality includes an equal sign (≤ or ≥), you'll use a solid line or curve. If it does not ( < or >), use a dashed line to indicate that points on the line or curve are not included.

Once the boundary is graphed, determine which side of it should be shaded. This represents all the points that satisfy the inequality. You can usually test this by picking a point not on the boundary (often the origin if it’s not on the boundary) and checking if it satisfies the inequality.

Circle Inequalities

For circle inequalities like the one in the exercise, the inequality is typically of the form: \(x^2 + y^2 \leq r^2\) where \(r\) is the radius of the circle. This inequality represents all the points inside or on the boundary of a circle with radius \(r\) and center at the origin (0,0).

Let's recap how to graph it:

• Start by drawing the circle with the given radius.
• Since the inequality symbol is ≤, you'll shade the entire area inside the circle including the boundary.

In our exercise, \(r = 3\) so we draw a circle centered at the origin with a radius of 3 units. Then, shading the area inside the circle indicates all solutions to \(x^2 + y^2 \leq 9\).

Linear Inequalities

Graphing linear inequalities follows similar steps with a few specific considerations. For the inequality in our exercise, \(x + y \geq 3\), you'll first graph its boundary line by converting the inequality to an equation: \(x + y = 3\). This step helps you graph the straight line accurately.

Some key steps:

• Draw the boundary line. Here, the line has a slope of -1 and y-intercept of 3. This means it crosses the y-axis at (0, 3) and the x-axis at (3, 0). Because the inequality uses ≥, draw a solid line.
• Next, choose a point to determine which side of the line to shade. A good test point is (0,0). Plugging this into the inequality, 0 + 0 \geq 3 does not hold true, which means you shade the region above the line, where the inequality is satisfied.

Combining this with the circle's graph, the overlapping shaded area satisfying both conditions is your solution set.