Problem 22 - Elementary Algebra

Question

Graph the inequality \(y+3>1\).

Step-by-Step Solution

Answer

Question: Graph the inequality \(y+3>1\) and describe the steps involved in solving it. Answer: To graph the inequality \(y+3>1\), follow these steps: 1. Rewrite the inequality by isolating \(y\) (subtract 3 from both sides): \(y > -2\). 2. Draw a dashed line on the coordinate plane representing \(y = -2\) (since points on the line are not included in the solution). 3. Shade the region above the dashed line, as it contains the points where \(y > -2\). The graph represents all points that satisfy the given inequality, \(y+3>1\).

Step 1: Rewrite the inequality

To find the inequality in terms of \(y\), subtract 3 from both sides of the inequality: \(y+3>1 \Rightarrow y>1-3\).

Step 2: Solve for y

Now, we evaluate the expression on the right-hand side of the inequality: \(y>-2\) Our inequality is now \(y > -2\).

Step 3: Draw the graph of y = -2

The inequality \(y>-2\) represents all the points above the line \(y=-2\). To graph this inequality, first draw the line \(y=-2\) on the coordinate plane. Since the inequality is a "greater than" relation, we will draw a dashed line instead of a solid line, which indicates that the points on the line are not included in the solution.

Step 4: Shade the region above the line

Since we are looking for the points where \(y>-2\), we will shade the region above the dashed line to represent the solution for this inequality. All points in this shaded region will satisfy the given inequality. Now the inequality \(y+3>1\) is successfully graphed.

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is like a map where you can locate points using two numbers, called coordinates. It consists of two number lines that intersect: the x-axis, which runs horizontally, and the y-axis, which runs vertically. These lines divide the plane into four quadrants. Each point on the plane is represented by an ordered pair \(x, y\).

• The x-coordinate shows the position of the point horizontally, either to the left or right of the y-axis.
• The y-coordinate shows the position vertically, either up or down from the x-axis.

When graphing equations or inequalities, you plot points on this plane to visually represent the solutions.

Graphing with a Dashed Line

When graphing linear inequalities, such as \(y > -2\), the type of line you use matters. A dashed line is used to graph inequalities that do not include the boundary line itself. For instance, when you have a "greater than" (>) or "less than" (<) symbol.

A dashed line signifies that points lying directly on the line do not satisfy the inequality. You draw a dashed line in the same way as a solid line, starting by plotting a series of points that satisfy the equivalent equation (e.g., \(y = -2\)), then you connect these points with a dashed line.

This visual cue helps to easily separate points that are part of the solution from those that aren't.

Shading Regions for Inequalities

Shading is an important part of graphing inequalities. It visually represents all the solutions to the inequality. For example, with the inequality \(y > -2\), you shade the area above the line \(y = -2\). This indicates where y-values are greater than -2.

• To determine which side to shade, pick a test point that is not on the boundary line and substitute it into the inequality.
• If the inequality holds true, the region containing the point is the solution area, and you shade it.

Shading effectively displays all potential solutions to the inequality, making it easier to understand which values of y satisfy the given condition.