Problem 23 - Precalculus: Functions and Graphs

Question

Sketch the graph of the system of Inequalities. $$\left\\{\begin{array}{r}x^{2}+y^{2} \leq 4 \\\x+y \geq 1\end{array}\right.$$

Step-by-Step Solution

Answer

The solution is the region inside the circle in the upper half-plane above the line x + y = 1.

Step 1: Understand the Inequalities

We have two inequalities: \( x^2 + y^2 \leq 4 \) and \( x + y \geq 1 \). The first represents a circle centered at the origin with radius 2, and the second represents a region above the line \( x + y = 1 \).

Step 2: Graph the Circle

Graph the inequality \( x^2 + y^2 \leq 4 \). This inequality represents all the points within and on the circle centered at the origin (0,0) with radius 2. The circle equation \( x^2 + y^2 = 4 \) forms the boundary, and you shade the interior of the circle since \( \leq \).

Step 3: Graph the Line

Graph the line \( x + y = 1 \). Draw the line based on points such as where it intercepts the y-axis at (0,1) and the x-axis at (1,0). Shade the region above this line since we're looking for \( x+y \geq 1 \).

Step 4: Identify the Intersection Area

Identify the region that satisfies both inequalities. This will be the overlapping shaded region from the circle and the area above the line. Highlight this intersection area on the graph.

Key Concepts

Graphing Inequalities

When you graph inequalities, you're visualizing all the solutions for an inequality on a coordinate plane. It’s like painting a picture that tells you where the solutions can be. Think of it as shading the part of the graph that meets the conditions set by your inequality.
- First, you will solve and graph the equality part. - For example, if you have an equation like \( y = 2x + 3 \), you would graph it as a line.
- Next, decide which side of the line represents your inequality. For \( y \leq 2x + 3 \), you shade below the line, because it fulfills the condition of being less than or equal to.
- Always remember: use a solid line for \( \leq \) or \( \geq \), and a dashed line for \( < \) or \( > \). This shows whether points on the line are included or not in the solution set.

Linear Inequalities

Linear inequalities look a lot like regular linear equations, but instead of just an equals sign, they have inequality symbols like \(<, \leq, >,\) or \(\geq\). They can represent a range of values that satisfy an equation.
- The inequality \( x + y \geq 1 \) represents all points on the graph where the sum of \( x \) and \( y \) is at least 1.
- To graph it, transform it to \( y \geq -x + 1 \) to easily identify the slope and y-intercept.
- Plot the line using the intercepts, and since it's a \( \geq \) inequality, shade the region above the line. This captures all points where the inequality holds true.
- By doing so, you visualize every potential solution on the grid which is seen as the shaded area.

Circle Inequality

Circle inequalities involve expressions like \( x^2 + y^2 \leq r^2 \), which means you're dealing with a circle in the coordinate plane. The equation \( x^2 + y^2 = 4 \) represents a circle centered at the origin (0,0) with a radius of 2. But the inequality \( x^2 + y^2 \leq 4 \) includes all the points inside and on the circle.
- To graph this, plot the circle using the circle equation, \( x^2 + y^2 = 4 \), as your boundary.
- Shade the interior of the circle to show all points that satisfy \( \leq 4 \). These include both points on the circle's edge and those inside it.
- Circle inequalities often represent bounded regions, emphasizing all values within a specific distance from a central point. Here, it means finding where solutions live relative to that bounded circular area. This shaded region reflects the inequality's influence.