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### Lecture 4

1. 1. Lecture 4<br />Linear Inequalities and Absolute Value Inequalities<br />
2. 2. Solving InequalitiesIntroduction<br /> In this course you will also need to be able to use the mathematical shorthand which describes a statement such as  "I'm talking about all numbers > than 4." <br /> We have special symbols to represent such statements, which we call "inequalities."<br /> For example, if we write " x > 4," we have described all numbers that are greater than 4.<br /> Similarly, if we write " 0 < x < 1," we have described all numbers that are between 0 and 1, but not including 0 and 1.<br />
3. 3. Solving InequalitiesIntroduction<br />Question: How many numbers are there between 0 and 1? <br />I hope you said "there's a bunch!"  <br />In fact, there are infinitely many numbers between 0 and 1.  <br />We represent this fact with a "number line graph" as follows: <br />
4. 4. Solving Inequalities Explanation<br /> In this section we will focus on problems which involve "solving inequalities." <br /> As in solving equations, "to solve" means to isolate the variable term so as to determine all values of the variable which will make the original statement true.<br /> In most cases, when we solve an inequality, our answer will consist of a piece (or pieces) of the real number line, rather than a few specific number values. <br /> For example, if we solve the equation  x + 5 = 8<br /> We get the number x = 3 as the only solution. <br />
5. 5. Solving Inequalities Explanation<br /> However, if we solve the inequality x + 5 > 8<br /> we get the "piece of the number line" described <br /> by x > 3 as our solution. <br /> That is, <br />
6. 6. Special Notation<br /> Inequality problems involve the following special symbols and notation, read from left to right.   <br />   means "less than or equal to"<br />   means "greater than or equal to"<br />   means "strictly less than"<br />   means "strictly greater than"<br />Solutions to inequality problems can be written in three  different ways. <br />
7. 7. Special Notation<br />For example, the statement "all numbers less than or equal to 5“ can be written using <br />1)"inequality form“ <br />2)a "number line graph“ <br />Notes:<br /> 1. The included endpointx = 5 is indicated with a "solid dot".              <br /> 2. An "open dot" is used when an endpoint is not included. <br />3) "interval form“ <br />Notes:<br /> 1. The included endpointx = 5 is indicated with a "bracket".  <br />    2. Parentheses are used when an endpoint is not  included. <br />    3. The symbol  represents "negative infinity." <br />
8. 8. Special Notation - continued<br />Another example is the statement  <br />"numbers which are less than - 1 or greater than or equal to 2."   <br />1) "inequality form"<br />  or  <br /> <br />2) a "number line graph"<br /> <br />3)  "interval form"<br />
9. 9. Vocabulary Notes<br />1.  An interval which includes both endpointsis called a "closed interval." <br />                                         <br />For example, [2,3]. <br />                                     <br />2.  An interval which does not include eitherendpointis called an "open interval." <br />                                          <br /> For example, (1,0).<br />
10. 10. Basic Solving Rules for Inequalities<br />When solving inequality problems, we will be able to use <br />many of the same rules we've already used when solving <br />equations. <br />Remember, our goal is to perform a sequence of <br />operations so we can isolate the desired variable term. <br />In order to isolate the variable in an inequality, we must <br />Remember two new solving rules.<br />
11. 11. Basic Solving Rules for Inequalities<br />New Rule 1:  <br />When you multiply or divide by a negative number, <br />you must reverse the direction of the inequality. <br />For example:  <br />solving <br />                      <br />leads to  <br />                      <br />so we get as our solution.<br />
12. 12. Basic Solving Rules for Inequalities<br />New Rule 2:  <br />Unlike solving equations, clearing out a variable <br />denominator in an inequality problem introduces <br />possible confusion over whether to reverse the direction of <br />the inequality. <br />Therefore, we will never multiply or divide by a variable <br />term, because it will be too hard to keep track of reversals in <br />the inequality.  <br />Instead we will construct a special number line which will <br />help us keep track of all possible outcomes in the problem.<br />
13. 13. Linear Inequalities<br />Linear inequalitiesare comprised of variable expressions in which <br />the variable only occurs to the first power. Also, no variable <br />denominators occur. <br />For example <br />Remember that linear equations can be solved using only the rules <br />of arithmetic.  <br />
14. 14. Linear Inequalities<br />The same is true for linear inequalities, but we also need to <br />remember the new solving rule: <br />When you multiply or divide by a negative number,<br />you must reverse the direction of the inequality.<br />
15. 15. Linear Inequalities - Example<br />For example, to solve the linear inequality <br />multiply each term by 2 <br />
16. 16. Linear Inequalities - Example<br />subtract "12x" from both sides <br />subtract 8 from both sides <br />divide both sides by - 11, <br />remembering to reverse the direction of the inequality<br />
17. 17. Linear Inequalities - Example<br />So our solution is:<br />