Problem 73 - Algebra 1

Question

Sketch the graph of the inequality in a coordinate plane. $$ y>-2 $$

Step-by-Step Solution

Answer

The graph is a dashed horizontal line at \(y = -2\) with the area above the line shaded indicating \(y > -2\).

Step 1: Draw the Line

The first step is to draw the line \(y = -2\) in a coordinate plane. Start with the y-intercept of -2, then since the slope is 0 (because there's no \(x\) in the equation), draw a horizontal line through this point.

Step 2: Indicate the Solution Region

Since the inequality sign is 'greater than', shade the region above the line. This also implies that the line \(y = -2\) is not included in the solution, hence the line should be dashed to indicate this fact.

Step 3: Interpretation

The shaded part of the graph represents all the points \((x, y)\) that satisfy the inequality \(y > -2\). That means for these points, the y-coordinate is greater than -2.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical coordinates. These numbers correspond to the distances from two intersecting lines, known as axes, which divide the plane into four quadrants. To visualize this in our day-to-day life, imagine a city grid, where streets act as lines that guide you to specific locations. In terms of mathematics, the horizontal line is called the x-axis, while the vertical line is the y-axis.

The origin, where these axes intersect, is the starting point (0,0). Negative and positive numbers on these axes help in locating points. For example, when graphing an equation or an inequality, the coordinate plane allows us to represent this visually, transforming an abstract concept into a tangible illustration that's easier to understand. This visual aid serves as a foundational tool for understanding more complex mathematical topics.

Inequality Notation

Understanding inequality notation is crucial to interpreting math problems involving comparisons. Inequality symbols include 'greater than' (>), 'less than' (<), 'greater than or equal to' (≥), and 'less than or equal to' (≤). Each symbol sets the relationship between two values or expressions.

For example, the inequality y > -2 tells us that y is any value greater than -2. It's an open inequality, meaning the number -2 is not part of the solution. When graphing, this is represented by a dashed line to show that the line itself is excluded from the set of solutions. Conversely, if the inequality included -2 (y ≥ -2), it would be graphed with a solid line, indicating that -2 is included. This notation is a compact way to represent a range of answers, providing a quick and clear method to convey mathematical ideas.

Solution Region in Graph

In graphing inequalities on a coordinate plane, the 'solution region' is the area that contains all the points that satisfy the inequality. It's like casting a net over the section of the plane where our conditions hold true. For the inequality y > -2, the solution region is above the line y = -2, as it includes all points where the y-value is greater than -2.

To indicate this on a graph, we shade the entire area above this line. Furthermore, because the inequality is strict (not including the line itself), we use a dashed line to graph y = -2. This visually informs anyone reading the graph that the line is the boundary that isn't part of the solution set. Essentially, the solution region can be thought of as the 'answer area' on the graph, where any point within this region is a valid solution to the inequality.