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Inequalities One Variable (Number Line)

Inequalities refer to problems that have any of these four symbols instead of an equal sign. Less than (<) Greater than (>) Less than or equal to (  ) Greater than or equal to (  ) Definition of an Inequality

x < 4 x  -2 x > 0 x  2 Examples of Inequalities:

Graphing Inequalities: x > 3 The simplest method for graphing these is to imagine the inequality is an arrow head. For example, treat the inequality > as if it is an arrow pointing to the right. This tells you which direction to go when drawing your shaded arrow on the number line. For this example, start at the number 3, called the “endpoint” and shade to the right

Here are the graphs for the four inequalities on the previous slide x < 4 x  -2 x > 0 x  2

2. The circle at the endpoint is open when it is (greater than) (these are called strict inequalities) x  x < 4 3. The circle at the endpoint is closed when it is either  (less than or equal to) or  (greater than or equal to) You should notice a few things about these graphs: 1. The variable “x” is always on the left side of the inequality.

There are also Inequalities which use two (or more) endpoints. An Example: x -1 It’s graph: Shade left starting at 2. Endpoint is an open circle Shade right starting at -1. Endpoint is an open circle “and” means both statements must be true. i.e. those numbers in the shaded area must be less than 2 AND they also have to be greater than -1.

Inequalities that use two (or more) endpoints. Another Example: x  3 or x < 0 It’s graph: Shade right starting at 3. Endpoint is a closed circle Shade left starting at 0. Endpoint is an open circle “or” means either statement can be true. i.e. Those numbers in the shaded area are either less than 0 or greater than 3.

Inequalities with use two (or more) endpoints. Another Example: -2 < x < 3 It’s graph: x lies BETWEEN the two endpoints. -2 on the left, 3 on the right. So the “shading”, representing the x, is also BETWEEN the two endpoints. Open circles on both sides.

EndpointEndpoint Shading Direction This chart tells you which way to shade and whether you use a solid dot or an open dot at the endpoint. left right closed circle   open circle

The Inequality The Graph Set Notation Interval Notation x < 5 x > -2 x  0 x  {x|x<5} (- ,5) {x| x > -2} (-2,  ) {x| x  -1} [-1,  ) {x| x  0} (- ,0] Examples including Set and Interval Notation

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