ࡱ> IKH5@ Zebjbj22 ,tXX(    f(f(f(8(t){),: - - -ii izzzzzzz$|R<{MjhiMjMj{   - -;{oooMj V - -zoMjzooo:t,`Ku -) %f( kt cy{0{tRmPKud^@    Kuiioi ipiii{{d"doo:"Unit 3.1 Linear Inequalities in One Variable Inequalities are algebraic expressions related by is less than, is less than or equal to, etc. Graphing inequalities on a number line: graph the interval on the number line that represents all the numbers that satisfy the inequality and use interval notation to write the interval. For example: Examples of open intervals: SetInterval NotationGraph{x | a < x} {x | a < x < b} {x | x < b} {x | x is a real #}(a, ") (a, b) (-", a) (-", ")  Examples of half-open intervals: SetInterval NotationGraph{x | a d" x} {x | a d" x < b} {x | x d" a}[a, ") [a, b) (-", a]  Examples of closed interval: SetInterval NotationGraph{x | a d" x d" b}[a, b] A linear inequality in one variable:  EMBED Equation.3  where A, B, and C are real numbers and A `" 0. Solving linear inequalities requires the same properties as solving linear equations. Solving linear inequalities with the addition property: For all real numbers A, B, and C, the inequalities A < B and A + C < B + C are equivalent. In other words, you can add the same number to both sides of an inequality and not change the solution set. Examples:  EMBED Equation.3  Solve and show solution set as interval notation and a graph.  EMBED Equation.3  Solve and show solution set as interval notation and a graph. Note: Rewrite inequalities as necessary so that the variable is on the left. In other words, rewrite 14 d" m as m e" 14. This will help avoid errors. Solving linear inequalities using the multiplication property: What happens when you multiply both sides of the true statement -2 < 5 by 8? You get another true statement -16 < 40. What happens when you multiply both sides of the true statement -2 < 5 by -8? You get a false statement 16 < -40! To make it a true statement you must reverse the inequality symbol. Multiplication Property of Inequality For all real numbers A, B, and C, with C `" 0 a) the inequalities A < B and AC < BC are equivalent if C > 0 ; b) the inequalities A < B and AC > BC are equivalent if C < 0 . In other words, each side of an inequality may be multiplied (or divided) by a positive number without changing the direction of the inequality symbol, but multiplying (or dividing) by a negative number requires we reverse the inequality symbol. Examples:  EMBED Equation.3  Solve and show solution set as interval notation and a graph.  EMBED Equation.3  Solve and show solution set as interval notation and a graph. Solving a linear inequality. Simplify each side separately. Use the distributive property if necessary and combine like terms. Isolate the variable terms on one side of the inequality. Use the addition property. Isolate the variable. Use the multiplication property to isolate the variable. Remember to reverse the inequality symbol is multiplying (or dividing) by a negative number. Linear inequalities with three parts  EMBED Equation.3   EMBED Equation.3  In this case x+2 is between 3 and 8 (and doesnt include 3 or 8). Note: Be sure to write three part inequalities in the correct order. It is incorrect to write  EMBED Equation.3  as this implies that 8 is less than 3. To solve this three part inequality, we would subtract 2 from all three parts.  EMBED Equation.3  resulting in this  EMBED Equation.3 . The solution set is (1, 6). Solving applied problems using linear inequalities. Be able to interpret terms such as: Word ExpressionInterpretationa is less than ba < ba is greater than ba > ba is at least ba e" ba is no less than ba e" ba is at most ba d" ba is no more than ba d" b Unit 3.2  Set Operations and Compound Inequalities The intersection of two sets is defined using the word and. We are looking for elements that are members of both sets simultaneously.  Here we are looking for those elements that are in the green, i.e. members of set A and of set B.  EMBED Equation.DSMT4 {x | x is an element of A and x is an element of B}. Example: A = {1, 2, 3, 4} and B = {2, 4, 6} then  EMBED Equation.DSMT4  A compound inequality consists of two inequalities linked by a connective word such as and or or.  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4   EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  Solving a compound inequality with the word and. Solve each inequality separately. Since the inequalities are joined with and, the solution set includes all numbers that satisfy both inequalities. Union of Sets The union of two sets is defined using the word or. We are looking for elements that are members of either set (or both).   EMBED Equation.DSMT4 = {x | x is an element of A or x is an element of B}. Example: A = {1, 2, 3, 4} and B = {2, 4, 6} then  EMBED Equation.DSMT4  Solving a compound inequality with the word or. Solve each inequality separately. Since the inequalities are joined with or, the solution set includes all numbers that satisfy either inequality. Unit 3.3 Absolute Value Equations and Inequalities Use the distance definition of absolute value. The absolute value of a number, x, represents the distance from x to 0 on the number line. What are the solutions to the absolute value equation  EMBED Equation.3  ? 4 & -4 In set notation this is {-4, 4}  SHAPE \* MERGEFORMAT  What about  EMBED Equation.3  ? Use the distance definition of absolute value to describe the answer. All numbers that are more than 4 units from 0.  SHAPE \* MERGEFORMAT  In interval notation  EMBED Equation.3  Another way to look at this is x < -4 or x > 4. What about  EMBED Equation.3  ? Use the distance definition of absolute value to describe the answer. All numbers that are less than 4 units from 0.  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Absolute value equations and inequalities take the form:  EMBED Equation.3  ,  EMBED Equation.3  , or  EMBED Equation.3 . (k is a positive number and a is not 0) The above examples show solution sets of:  EMBED Equation.3  has the same solution set as x = -4 or x =4.  EMBED Equation.3  has the same solution set as x < -4 or x > 4.  EMBED Equation.3  has the same solution set as x > -4 and x < 4 (-4 < x < 4). Solving Absolute Value Equations and Inequalities (k is a positive real number) To solve  EMBED Equation.3  , solve the compound equation  EMBED Equation.3  or  EMBED Equation.3 . The solution set is usually of the form {p, q}.  SHAPE \* MERGEFORMAT  To solve  EMBED Equation.3  , solve the compound inequality  EMBED Equation.3  or  EMBED Equation.3 . The solution set is of the form  EMBED Equation.3 .  SHAPE \* MERGEFORMAT  To solve  EMBED Equation.3  , solve the three part inequality  EMBED Equation.3 . The solution set is of the form  EMBED Equation.3 .  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