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Functions and Graphs Help (page 4)

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Updated on Oct 27, 2011

Example

    2x + y > 1
    Subtract 2x from both sides of the inequality. 2x – 2x + y > 1 – 2x  
    Simplify. y > 1 – 2x  
    Use the commutative property. y > –2x + 1

The inequality y > –2x + 1 tells you that the slope is –2 and the y-intercept is 1. If the inequality has a > or < symbol, then the line will be dotted. If the inequality symbol is ≤ or ≥, then the line will be solid. Generally, if the inequality symbol is > or ≥, you shade above the line. If the inequality symbol is < or ≤, you shade below the line.

To graph y > –2x + 1, start with the y-intercept, which is 1. The slope is –2, which means –, so from the y-intercept of 1, go down 2, because the slope is negative, and to the right 1. When you connect the starting point and the ending point, you will have the boundary line of your shaded area. You should extend this line as far as you'd like in either direction because it is endless. The boundary line will be dotted because the inequality symbol is >. If the symbol had been ≥, then the line would be solid, not dotted.

Checking Your Graph

When you graph, always check your graph to make certain the direction of the line is correct. If the slope is positive, the line should rise from left to right. If the slope is negative, the line should fall from left to right.

Functions and Graphs

Now that you have the boundary line, will you shade above or below the line? The inequality symbol is >, so shade above the line.

Functions and Graphs

If the inequality symbol is > or ≥, you will shade above the boundary line. If the inequality symbol is < or ≤, you will shade below the boundary line.

Special Cases of Inequalities

There are two special cases of inequalities. One has a vertical boundary line and the other has a horizontal boundary line. For example, the inequality x > 2 will have a vertical boundary line, and the inequality y > 2 will have a horizontal boundary line. The inequality y > 2 is the same as the inequality y > 0x + 2. It has a slope of 0 and a y-intercept of 2. When the slope is 0, the boundary line will always be a horizontal line.

Functions and Graphs

The inequality x > 2 cannot be written in y = form because it does not have a slope or a y-intercept. It will always be a vertical line. It will be a dotted vertical line passing through the point on the x-axis where x = 2.

Functions and Graphs

A horizontal line has a slope of 0. A vertical line has no slope.

Real World Inequality Graphs

When you graph an inequality, you have a picture of all possible answers to a problem. Can you see that it would be impossible to list them all? Think how valuable graphs are to the business world. If the shaded area on the graph represents all the possible prices a company can charge for a product and make a profit, executives can easily determine what to charge for a product to achieve the desired profit.

Find practice problems and solutions for these concepts at Functions and Graphs Practice Problems.

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