Solving Inequalities Chapter 3 3 1 Inequalities and

Solving Inequalities Chapter 3

3. 1 Inequalities and Their Graphs � Pg. 164 – 170 � Obj: Learn how to write, graph, and identify solutions of inequalities. � Content Standard: Prepares for A. REI. 3

3. 1 Inequalities and Their Graphs � Solution of an inequality – any number that makes the inequality true � Representing Inequalities ◦ < or > open circles ◦ < or > closed circles

3. 2 Solving Inequalities Using Addition or Subtraction � Pg. 171 – 177 � Obj: Learn how to use addition or subtraction to solve inequalities. � Content Standards: A. REI. 3 and A. CED. 1

3. 2 Solving Inequalities Using Addition or Subtraction � Equivalent Inequalities – inequalities that have the same solutions � Solving Addition or Subtraction Inequalities ◦ Isolate the variable ◦ Do the opposite operation �Addition – Subtraction �Subtraction – Addition ◦ Whatever you do to one side of the equation, you must do to the other

3. 3 Solving Inequalities Using Multiplication or Division � Pg. 178 – 183 � Obj: Learn how to use multiplication or division to solve inequalities. � Content Standards: A. CED. 1, N. Q. 2, and A. REI. 3

3. 3 Solving Inequalities Using Multiplication or Division � Solving Multiplication or Division Inequalities ◦ Isolate the Variable ◦ Do the opposite operation �Multiplication – Division �Division – Multiplication �If you multiply or divide by a negative, change the direction of the inequality. ◦ Whatever you do to one side, you must do to the other

3. 3 Solving Inequalities Using Multiplication or Division � Concept Byte – Pg. 184 � Properties of Equality ◦ Reflexive Property – a = a ◦ Symmetric Property – If a=b, then b=a ◦ Transitive Property – If a=b and b=c, then a=c � Transitive Property of Inequality ◦ If a<b and b<c, then a<c

3. 4 Solving Multi-Step Inequalities � Pg. 186 – 192 � Obj: Learn how to solve multi-step inequalities. � Content Standards: Prepares for A. REI. 3 and A. CED. 1

3. 4 Solving Multi-Step Inequalities � Solving Inequalities ◦ Use the Distributive Property to remove any grouping symbols ◦ Combine like terms on each side of the inequality ◦ Get the variable terms on one side of the inequality and the constants on the other ◦ Solve for the variable ◦ Check your solution in the original inequality

3. 5 Working With Sets � Pg. 194 – 199 � Obj: Learn how to write sets and identify subsets and find the complement of a set. � Content Standard: A. REI. 3

3. 5 Working With Sets � Roster Form – lists the elements of a set within braces {} – {2, 4, 6, 8, …} � Set-builder notation – describes the properties an element must have to be in included in a set – {x|x is a multiple of 2} – “the set of all real numbers x, such that x is a multiple of 2” � Empty Set – the set that contains no elements � Universal Set – the largest set you are using � Complement of a Set – the set of all elements in the universal set that are not in the set (A’)

3. 6 Compound Inequalities � Pg. 200 – 206 � Obj: Learn how to solve and graph inequalities containing the words “and” or “or”. � Content Standards: A. REI. 3, and A. CED. 1

3. 6 Compound Inequalities � Compound Inequality – consists of two distinct inequalities joined by the and or the word or � Interval notation ◦ Parentheses – Use (or) when a < or > symbol indicates that the interval’s endpoints are not included ◦ Brackets – Use [or] when a < or > symbol indicates that the interval’s endpoints are included ◦ Infinity – Use ∞ when the interval continues forever in a positive direction. Use -∞ when the interval continues forever in a negative direction.

3. 7 Absolute Value Equations and Inequalities � Pg. 207 – 213 � Obj: Learn how to solve equations and inequalities involving absolute value. � Content Standards: A. CED. 1, and A. SSE. 1. b

3. 7 Absolute Value Equations and Inequalities � Solving Absolute Value Equations � Solving Absolute Value Inequalities ◦ Set what is inside the absolute signs equal to both the positive and negative values ◦ Solve each equation and check your solutions ◦ Use method similar to absolute value equations ◦ < or < is an “and” compound inequality ◦ > or > is an “or” compound inequality

3. 8 Unions and Intersections of Sets � Pg. 214 – 220 � Obj: Learn how to find the unions and intersections of sets. � Content Standard: A. CED. 1

3. 8 Unions and Intersections of Sets � Union – the set that contains all elements of the sets (A ∪ B) � Intersection – the set of elements that are common to every set (A ∩ B) � Disjoint Sets – sets that have no elements in common