Problem 61 - Calculus with Concepts in Calculus

Question

Sketch the region in the plane satisfying the given conditions. \(x \geq-1\) and \(y \geq \frac{1}{2}\)

Step-by-Step Solution

Answer

Shade the area above the line \(y = \frac{1}{2}\) and to the right of the line \(x = -1\).

Step 1: Understand the Inequalities

We have two inequalities: \(x \geq -1\) and \(y \geq \frac{1}{2}\). These describe regions on the Cartesian plane. The inequality \(x \geq -1\) indicates that we are looking at all points to the right of the vertical line \(x = -1\). The inequality \(y \geq \frac{1}{2}\) indicates that we are looking above the horizontal line \(y = \frac{1}{2}\).

Step 2: Identify the Lines

For the inequality \(x \geq -1\), the boundary is the vertical line at \(x = -1\). For \(y \geq \frac{1}{2}\), the boundary is the horizontal line at \(y = \frac{1}{2}\). These lines form the borders of the region we are interested in.

Step 3: Sketch the Lines

Draw a coordinate plane. First, sketch the line \(x = -1\), which is a vertical line crossing the x-axis at -1. Then sketch the line \(y = \frac{1}{2}\), which is a horizontal line crossing the y-axis at 0.5.

Step 4: Shade the Region

Identify the region to the right of the line \(x = -1\) where \(x\) values are greater than -1, and above the line \(y = \frac{1}{2}\) where \(y\) values are greater than 0.5. This intersection forms a quadrant or a sector of the plane. Shade this area to represent the solution.

Step 5: Verify the Solution

Choose a test point such as \((0,1)\), which lies in the shaded region. Check if it satisfies both inequalities: \(0 \geq -1\) is true, and \(1 \geq \frac{1}{2}\) is also true, confirming that the region is correctly shaded.

Key Concepts

Understanding the Coordinate Plane

A coordinate plane is crucial for graphing equations and inequalities, providing a visual workspace for mathematical concepts. The plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical), creating four quadrants. Any point on this plane is defined by an ordered pair \(x, y\).
For instance, in the exercise, we consider points on the plane that satisfy certain conditions given by inequalities. These points are plotted relative to the boundary lines, helping to visually represent solution sets. This visual representation enables easier understanding and interpretation of mathematical solutions.

Solving Inequality Solutions

Inequality solutions on a coordinate plane are depicted by regions rather than discrete points. Given the inequalities \(x \geq -1\) and \(y \geq \frac{1}{2}\), the solutions are all the coordinates that satisfy both conditions.
Here's how inequality solutions are identified:

• Start by considering the inequality \(x \geq -1\). The solution consists of all points with \(x\) values that are greater than or equal to -1, forming a region to the right of the vertical line \(x = -1\).
• The inequality \(y \geq \frac{1}{2}\) includes all points with \(y\) values greater than or equal to 0.5, creating a region above the horizontal line \(y = \frac{1}{2}\).

Finally, the intersection of these regions marks the area of feasible solutions—visually represented by shading this section. It signifies the simultaneous satisfaction of both inequalities and lets us choose any point within this area as a valid solution.

Roles of Boundary Lines in Graphing

Boundary lines are essential in graphing inequalities, acting as markers that divide the plane into distinct regions. For the equations \(x = -1\) and \(y = \frac{1}{2}\), these lines create the boundaries used to define our region.
Here's how they work:

• For \(x \geq -1\), the boundary line runs vertically at \(x = -1\). This line serves as the limit beyond which all \(x\) values are to the right.
• In the case of \(y \geq \frac{1}{2}\), the boundary is the horizontal line \(y = \frac{1}{2}\), above which all \(y\) values satisfy the inequality.

The boundary lines are often drawn as solid lines if the inequality is inclusive (e.g., \(\geq\)), showing that points on the line are part of the solution. Understanding these lines helps in identifying which areas on the plane satisfy the given inequalities, ensuring correct graphical representation.