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