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Writing a linear inequality from a graph is similar to writing a linear equation from a graph. The biggest difference is that with inequalities, a portion of the coordinate plane is highlighted to represent the solution region. Another thing to consider is how the line is drawn: whether a solid or a dashed line is used. These two characteristics, the highlighted solution region and the type of line drawn, will help us determine what inequality symbol to use when writing the inequality.
First, we'll start with determining the other parts of the inequality. To do this, we will somewhat ignore the fact that we're dealing with an inequality, at least for now, and focus on the boundary line to write an equation in the form
EXAMPLE
Write the inequality that corresponds to the graph below:If a boundary line is a vertical line, this can be presented by the equation . The inequality that we will write to represent this graph has no y-component at all. This simply highlights all x-values on one side of the boundary line, no matter what the value of y is.
Once again, we look at the type of line used in the graph to determine if our inequality symbol will be strict or non-strict. However, we interpret "above the line" or "below the line" a bit differently. Think about the values of x that run along the x-axis. As we read the graph from left to right, our x-values go from negative infinity to positive infinity. Thus, "above the line" should be interpreted as "to the right" and "below the line" should be interpreted as "to the left."
EXAMPLE
Write the inequality that corresponds to the graph below:If a boundary line is a horizontal line, this can be presented by the equation . In contrast to our vertical boundary lines, this has no x-component in the equation or inequality. This means that all values either greater than or less than a certain value for y are in the solution region, no matter what the value of x is.
EXAMPLE
Write the inequality that corresponds to the graph below:Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License