Problem 20 - Algebra and Trigonometry

Question

\(19-32\) Sketch the region given by the set. $$ \\{(x, y) | y \geq 0\\} $$

Step-by-Step Solution

Answer

Sketch the region above and including the x-axis.

Step 1: Understand the Given Set

We are given the set \( \{(x, y) | y \geq 0 \} \). This inequality describes a region in the coordinate plane where the \( y \)-value of each point is greater than or equal to zero. Therefore, the region of interest is above or on the x-axis.

Step 2: Identify the Critical Boundary

The critical boundary for this inequality is the line \( y = 0 \), which is the x-axis itself. This means all points directly on the x-axis are included in our region since \( y \) can be equal to zero.

Step 3: Determine the Shading Direction

Since the set includes all points where \( y \) is greater than or equal to zero, we shade the entire upper half-plane above the x-axis. Make sure to include the x-axis in your shading because \( y = 0 \) is also part of the set.

Step 4: Sketch the Region

Draw the Cartesian plane with horizontal and vertical axes. Shade the region above the x-axis, including the x-axis itself. The shaded area represents all points that satisfy \( y \geq 0 \).

Key Concepts

Inequalities in Two Variables

Inequalities in two variables, such as the one given in the exercise, are equations that express relationships between two variables. Instead of showing a precise equal value, inequalities provide a range of potential values for one or both variables. In this context, the inequality is given by \( y \geq 0 \), which describes any point \((x, y) \) where the \( y \)-value is either zero or a positive number.

Understanding inequalities involves recognizing that instead of a single line or point, you have an entire region of solutions. The boundary line of the inequality \( y \geq 0 \) is \( y = 0 \), which is the x-axis. This line serves as a demarcation that separates the potential solution space from non-solution areas. Remember, the solutions include the boundary when the inequality symbol is "greater than or equal to" (\( \geq \)) or "less than or equal to" (\( \leq \)).

Graphing in Algebra

Graphing in algebra is all about translating mathematical equations or inequalities into visual representations on the coordinate plane. A coordinate plane consists of two perpendicular axes: x-axis and y-axis. Each point \((x, y)\) on this plane corresponds to a pair of coordinates from the x and y axes.

When graphing inequalities, the task is often about identifying areas on the plane that satisfy the equation rather than a distinct line or curve. For example, the inequality \( y \geq 0 \) involves graphing its boundary, which is the line \( y = 0 \). The entire space above (including) this line represents the solution area. When graphing inequalities in two variables, always reflect:

• What is the boundary line?
• Should this boundary be solid or dashed?
• Which side of this line satisfies the inequality?

By answering these, you can confidently map out the correct region that holds all possible solutions to the given inequality.

Shading Regions on Graphs

Shading regions on graphs is a visual technique used to represent all possible solutions of an inequality on the coordinate plane. For inequalities like \( y \geq 0 \), shading is a great way to show which parts of the plane make the inequality true. In this case, the area you would shade is above the x-axis, including the axis itself for points where \( y = 0 \).

When shading:

• First, draw the boundary line. For \( y \geq 0 \), this is the x-axis.
• Decide on the shading direction. Since it states \( y \geq 0 \), shade the area where \( y \) values are positive, i.e., the region above the x-axis.
• If the inequality includes an "equal to" aspect (like \( \geq \)), the boundary line itself is solid to show these points are included.

Using this approach can significantly help in verifying solutions and understanding the scope of the inequality visually. Shading is often the final step in graphing as it clearly communicates the solution set.