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Section 6.1 Solving Inequalities Pass Skills 2.3a Algebra I Obj: state and use symbols of inequality. Obj: solve inequalities that involve addition and subtraction. Vocab Addition property of inequality let a, b, and c be real #’s. If a <b, then a + c < b + c. Adding equal amounts to each side of an equation results in an equivalent equation. Inequality a statement that 2 expressions are not equal. Contain the following signs: <,>,,, Subtraction property of inequality let a, b, and c be real #’s. If a < b, then a – c < b – c. If equal amounts are subtracted from the expresseions on each side of an inequality the resulting inequality is true. Statements of Inequality a is less than b. a<b a is greater than b. a>b a is less than or equal to b. ab a is greater than or equal to b. ab a is greater than b and less than c. b<a<c a is greater than or equal to b and less than or equal to c. bac a is not equal to b. ab Solve inequalities, and graph the solution on a number line. x + 12 16 x + 12 -12 16 -12 x4 Section 6.2 Multistep Inequalities Algebra I Obj: state and apply the multiplication and division properties of inequality. Obj: solve multistep inequalities in one variable. Vocab Division property of inequality let a, b, and c be nonzero real #’s. For c > 0, if a > b, then a/c > b/c, and if a < b, then a/c > b/c. For c < 0, if a < b, then a/c > b/c and if a > b then a/c < b/c Multiplication property of inequality let a, b, and c be nonzero real #’s. For c > 0, if a > b, then ac > bc, and if a < b, then ac < bc. For c < 0, if a < b, then ac> bc, and if a > b, then ac < bc. Summary of Multiplication and Division Properties of Inequalities ACTION RESULT Multiply or divide by a positive # inequality sign stays the same Multiply or divide by a negative # inequality sign must be reversed Solve 18 – 2y > 2. 18 – 2y > 2 18 – 18 – 2y > 2 – 18 –2y > –16 -2y/-2 < -16/-2 (Reverse inequality sign. ) Y<8 Section 6.3 Compound Inequalities Algebra I Obj: graph the solution sets of compound inequalities. Obj: solve compound inequalities. Vocab Compound inequality 2 inequalities that are combined into one statement by the word and or or. Conjunction a compound inequality whose solution region is an intersection. Disjunction a compound inequality whose solution region is outside an intersection. Intersection (of graphs) the solution to a system of linear equalities or inequalities, consisting of the solutions common to each. Union the union of 2 sets consists of all elements from both sets. The logical relationship OR represents the union of sets. Solve and graph 6 < 9y 3 24. 6 < 9y 3 and 9y – 3 24 1< y and y 3 conjunction: graph lies within the endpoints Section 6.4 Absolute Value Functions Algebra I Pass Skills 2.2b,2.2e Obj: explore features of the absolute value function. Obj: explore basic transformations of the absolute value function. Vocab Absolute value the absolute value of a real # x is the distance from x to 0 on a # line; the symbol |x| means the absolute value of x. Absolute value function a function written in the form y = |x| or y = ABS(x) Line of reflection the line across which a graph is reflected. Parent function the most basic function of a family of functions, or the original function before a transformation is applied. Transformation a variation such as a stretch, reflection, or translation of a parent function. Translation a transformation that shifts the graph of a function horizontally or vertically. Reflection a transformation that creates a mirror image of a given function. Rules for absolute value. |a| = a, for a 0 |-a| = a, for a 0 |a| = –a, for a < 0 Find the absolute value of an expression. Find |9 – 2|. |9 – 2| = |7| =7 Find |2 – 9|. |2 – 9| = |– 7| =7 Find the domain and range of y = 4|x|. domain: all real numbers range: all non-negative numbers Section 6.5 Absolute Value Equations and Inequalities Algebra I Pass skills 2.2e Obj: solve absolute value equations. Obj: solve absolute value inequalities and express the solution as a range of values on a number line. Vocab Absolute error the absolute value of the difference between the actual measure and the specified measure. Absolute value equation an equation that includes an absolute value; it will have 2 solutions. Absolute value inequality an inequality that includes an absolute value. Error the difference between the actual measure and the specified measure. Solve absolute-value equations. Solve |x + 3|= 7. Case 1: Case 2: x + 3 is positive x + 3 is negative x+3=7 x + 3 = –7 x =4 x = –10 solutions: x = –10 and x = 4 Solve absolute-value inequalities. Solve |x – 4| 5. Case 1: Case 2: x – 4 is positive x – 4 is negative x–45 x – 4 –5 x9 x –1 solution: –1 x 9