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Chapter 3: Problem 27

Graph the inequality. $$ y \leq 0 $$

Short Answer

Expert verified
Draw the x-axis as a solid line and shade below it.

Step by step solution

01

Understand the Inequality

The given inequality is \(y \leq 0\). This means that y can be any value that is less than or equal to 0.
02

Identify the Boundary Line

The boundary line for the inequality \(y \leq 0\) is the line \(y = 0\). This line is a horizontal line crossing the y-axis at 0.
03

Draw the Boundary Line

On the coordinate plane, draw a horizontal line through the point where y = 0. This line is the x-axis.
04

Shading the Correct Region

Since the inequality is \(y \leq 0\), shade the region below the boundary line (the x-axis). This represents all points where y is less than or equal to 0.
05

Conclude the Graph

The final graph should show the x-axis (y = 0) as a solid line with the area below it shaded to indicate all points where \(y \leq 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Plane
To start graphing any inequality, you need to understand the coordinate plane. A coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical coordinates, typically written as \((x, y)\). The horizontal axis is known as the x-axis, and the vertical one is the y-axis.
The point where these axes intersect is called the origin, denoted as \((0, 0)\).
When plotting points or drawing graphs, you will often move vertically and horizontally from this origin point.
Boundary Line
The boundary line is crucial for graphing inequalities. It separates the coordinate plane into two regions. In this exercise, the inequality is \(\text{y} \leq 0\), which means our boundary line is where y equals 0.
This particular boundary is a horizontal line because y stays constant along this line. To draw it:
  • Locate 0 on the y-axis.
  • Draw a straight line horizontally across the entire plane.
    For inequalities with less than or equal (\
Shading Regions
After drawing the boundary line, we determine which side of the line to shade. Shading shows all the solutions to the inequality. For \(\text{y} \leq 0\), you shade the region below the boundary line.
Why? Because \(\text{y} \leq 0\) means y can be any value less than or equal to 0.
To correctly shade the region:
  • Start at the boundary line,
  • Move downwards, and
  • Shade everything below the line.
    This shaded area represents all the points (x, y) that satisfy the inequality.

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