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1、Linear Inequalities,<,<,<,,,,,,,,,,,,, 2002 by Shawna Haider,There are two kinds of notation for graphs of inequalities: open circle or filled in circle notation and interval notation brackets. You should be familiar with both.,,,,,Both of these number lines show the inequality above. They are just

2、using two different notations. Because the inequality is greater than or equal to the solution can equal the endpoint. That is why the circle is filled in. With interval notation brackets, a square bracket means it can equal the endpoint.,,,circle filled in,squared end bracket,Remember---these mean

3、the same thing---just two different notations.,,Lets look at the two different notations with a different inequality sign.,,,),,Since this says less than we make the arrow go the other way. Since it doesnt say or equal to the solution cannot equal the endpoint. That is why the circle is not filled i

4、n. With interval notation brackets, a rounded bracket means it cannot equal the endpoint.,,,circle not filled in,rounded end bracket,Remember---these mean the same thing---just two different notations.,,Compound Inequalities Lets consider a double inequality (having two inequality signs).,,(,I

5、think of these as the inbetweeners. x is inbetween the two numbers. This is an and inequality which means both parts must be true. It says that x is greater than 2 and x is less than or equal to 3.,,,,,Compound Inequalities Now lets look at another form of a double inequality (having two i

6、nequality signs).,,),Instead of and, these are or problems. One part or the other part must be true (but not necessarily both). Either x is less than 2 or x is greater than or equal to 3. In this case both parts cannot be true at the same time since a number cant be less than 2 and also greater than

7、 3.,,,,,,,Just like graphically there are two different notations, when you write your answers you can use inequality notation or interval notation. Again you should be familiar with both.,,,0,4,6,8,2,,,,,Lets have a look at the interval notation.,For interval notation you list the smallest x can be

8、, a comma, and then the largest x can be so solutions are anything that falls between the smallest and largest.,,,,,,,,,,,,,,,,,unbounded,The bracket before the 1 is square because this is greater than or equal to (solution can equal the endpoint). The bracket after the infinity sign is rounded beca

9、use the interval goes on forever (unbounded) and since infinity is not a number, it doesnt equal the endpoint (there is no endpoint).,Lets try another one.,The brackets used in the interval notation above are the same ones used when you graph this.,(,,,This means everything between 2 and 4 but not i

10、ncluding -2,,,Lets look at another one,Notice how the bracket notation for graphing corresponds to the brackets in interval notation. Remember that square is or equal to and round is up to but not equal. By the infinity sign it will always be round because it cant equal infinity (that is not a numbe

11、r).,,,,,),,Now lets look an or compound inequality,There are two intervals to list when you list in interval notation.,,,When the solution consists of more than one interval, we join them with a union sign.,,,Properties of Inequalities.,Essentially, all of the properties that you learned to solve li

12、near equations apply to solving linear inequalities with the exception that if you multiply or divide by a negative you must reverse the inequality sign.,So to solve an inequality just do the same steps as with an equality to get the variable alone but if in the process you multiply or divide by a negative let it ring an alarm in your brain that says Oh yeah, I have to turn the sign the other way to keep it true.,Example:,- 4x - 4x,+ 6 +6,Ring the alarm! We divided by a negative!,