Problem 10 Sketch the graph of the solution... [FREE SOLUTION] (2024)

Short Answer

Boundary Line

When graphing a linear inequality, the first step is to rewrite the inequality as an equation to find the boundary line.
In our example, we start with the inequality: \(x - 2y \geq 6\).
We rewrite this as the equation: \[ x - 2y = 6 \].
The boundary line helps us visually separate the coordinate plane into different regions where the inequality may or may not hold.
Since the inequality symbol is greater than or equal to (\( \geq \)), it means that the points on the boundary line itself are also solutions.
We depict this with a solid line.
A dashed line would be used if the inequality was strictly greater than (\(>\)) or strictly less than (\(<\)).

Intercepts

Shading Regions

The shaded region on the graph represents all the solutions to the inequality.
After plotting the boundary line using the intercepts, you need to determine which side of the line contains the solutions.
Choose a test point that is not on the boundary line, like the origin (0, 0), to test the inequality.
In our example, substituting (0, 0) into \[ x - 2y \geq 6 \]:
\[ 0 - 2(0) \geq 6 \]
Simplifies to \[ 0 \geq 6 \], which is false.
Therefore, the region that does not contain the point (0, 0) is the solution region.
This region should be shaded to indicate all possible solutions to the inequality.

Solution Set

The solution set of the inequality is all the points (x, y) in the shaded region and on the boundary line.
These points satisfy the inequality \[ x - 2y \geq 6 \].
The shading gives a visual representation of the range of solution possibilities.
It includes every point below and to the right of the solid boundary line, confirming that all these points make the inequality true.
This set helps us understand and confirm our solutions visually.
By accurately drawing and shading the graph, we ensure that we capture all potential solutions to the given linear inequality.