Question

Graph each inequality. $$y \leq 2$$

Short answer

Draw a solid line at \( y = 2 \) and shade below it.

Step-by-step solution

Understand the Inequality

The inequality given is \( y \leq 2 \). This means that the value of \( y \) should be less than or equal to 2.

Draw the Boundary Line

To graph \( y \leq 2 \), first draw the boundary line for \( y = 2 \). This is a horizontal line that passes through the \( y \text{-axis} \) at \( y = 2 \). Because the inequality includes \( \leq \), use a solid line.

Shade the Solution Area

Since the inequality is \( y \leq 2 \), shade the region below the line. This region represents all the points \( (x, y) \) where \( y \) is less than or equal to 2.

Label the Graph

Ensure that the boundary line is labeled as \( y = 2 \) and the shaded region is clearly marked to represent \( y \leq 2 \). This confirms the solution area on the graph.

Key concepts

boundary line

In graphing inequalities, the boundary line is a crucial concept. For the inequality given, which is \( y \leq 2 \), the boundary line is \( y = 2 \).
To plot this boundary, you draw a horizontal line that passes through the point where \( y = 2 \) on the y-axis.
Since the inequality includes \( \leq \) (less than or equal to), you will use a solid line to indicate that points on the line are included in the solution set.
Remember, if the inequality were strict (\( < \) or \( > \)), you would use a dashed line instead to show points on the line are not part of the solution.

shaded region

The shaded region represents all possible solutions to the inequality.
After drawing the boundary line for \( y = 2 \), you need to determine which side of the line to shade.
For the inequality \( y \leq 2 \), the region below the boundary line is shaded.
This is because in this region, all points (x, y) have y-values less than or equal to 2.
Always ensure the shaded area is clearly marked. This visualization helps in understanding which values satisfy the inequality.

horizontal line

A horizontal line is a straight line that runs left to right and has the same y-value across all points.
In the inequality \( y \leq 2 \), the boundary line \( y = 2 \) is a horizontal line.
This means every point on this line has a y-coordinate of 2.
When drawing horizontal lines, make sure they are perfectly straight and extend across the graph for clarity.
Horizontal lines are simple but fundamental elements when graphing inequalities involving only one variable.

inequality graphing

Inequality graphing involves plotting regions on a graph where a given inequality holds true.
For \( y \leq 2 \), start by identifying the boundary line, which is \( y = 2 \).
This boundary separates the graph into two parts: where \( y \) is less than or equal to 2, and where \( y \) is greater than 2.
Next, shade the correct region - below the line for \( y \leq 2 \) - to indicate all solutions that satisfy the inequality.
Labeling the graph appropriately helps in visualizing and understanding the solutions correctly.