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Sets and Compound Inequalities Introduction to Sets A set is a collection of objects which are called elements. The elements are distinct and, thus, are listed only once. One way to identify a set is by the List or Roster Method, whereby the elements are listed within braces {}. To indicate that an object is an element of a set, the symbol ∈ is used. To indicate that an object is not an element of a set, the symbol ∉ is used. Example 1: Write the first four months of a year as set M using the List Method, identify one of the months as an element of the set, and identify a month that is not an element of the set. M = {January, February, March, April} February ∈ M July ∉ M A set that has no elements is called an empty set and is denoted by the symbol Ø. Sets can be joined together using the intersection of sets or the union of sets. The intersection of two sets A and B is the set of all elements that are common to both A and B and is denoted as A∩B The union of two sets A and B is the set of all elements in both A and/or B and is denoted as A∪B Example 2: Find the intersection: {2, 3, 4, 5, 6, 7} ∩ {1, 2, 5, 7, 8, 9} {2, 5, 7} Example 3: Find the union: {2, 4, 6, 8} ∪ {3, 5, 7} {2, 3, 4, 5, 6, 7, 8} Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Compound Inequalities Similar to sets, several inequalities can be combined together to form what are called compound inequalities. One type of compound inequality is an OR inequality. This type of inequality will result in a true statement from either one inequality OR the other inequality OR both. This compound inequality is called a disjunction. The solution set of a disjunction is the union of the solution sets of the individual inequalities. A convenient way to graph a disjunction is to graph each individual inequality above the number line, then move them both onto the actual number line. When writing the solution in interval notation, if there are two different parts to the graph, a ∪ (union) symbol is used between the two sets. Example 4: Graph and write interval notation for the disjunction: x > 4 or x < -2 (-∞, -2] ∪ (4, ∞) Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Example 5: Solve, graph, and write interval notation: 2x + 7 < -11 or -3x – 2 < 13 -7 -7 +2 +2 2x < -18 -3x < 15 2 2 -3 -3 x < -9 x > -5 Solve each inequality Subtract or add Divide Dividing by a negative flips the inequality sign Notice when graphing, an open dot was used rather than the parentheses. Unless indicated, either can be used. (-∞, -9) ∪ (-5, ∞) Example 6: Solve, graph, and write interval notation: x – 9 < -8 or 2x + 3 > -3 +9 +9 -3 -3 x <1 2x > -6 2 2 x > -3 Solve each inequality Add or subtract Divide When the graphs are combined, the solution covers the entire number line, so the interval notation is (-∞, ∞). Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) The second type of compound is an AND inequality. AND inequalities require both statements to be true. If one is false, they both are false. This compound inequality is called a conjunction. The solution set of a conjunction is the intersection of the solution sets of the individual inequalities. When graphing these inequalities, follow a similar process as with as above – first graph both inequalities above the number line, but where the two inequalities overlap will be drawn onto the number line for the final graph. When writing the solution in interval notation, a ∩ (intersection) symbol could be used but is not necessary and not commonly used. Example 7: Graph and write interval notation for the conjunction: x > 2 and x < 5 (2, 5] Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Example 8: Solve, graph, and write interval notation: 2x + 8 > 5x – 7 and 5x – 3 > 3x + 1 -5x - 5x -3x -3x -3x + 8 > - 7 2x – 3 > 1 -8 -8 +3 +3 -3x < -15 2x > 4 -3 -3 2 2 x < 5 x> 2 Solve each inequality Move variable to one side Subtract or add Divide Dividing by a negative flips the inequality sign (2, 5] Example 9: Solve, graph, and write interval notation: x + 7 < 10 – 4 and 5x – 7 > 3 x+7<6 5x – 7 > 3 -7 -7 +7+7 x < -1 5x > 10 5 5 Solve each inequality Combine like terms Subtract or add Divide There is no overlap of the two inequalities, so there is no solution or Ø. Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Conjunctions may also be expressed as a single sentence with the variable (or expression containing the variable) between two numbers. When solving these conjunctions, because there are three parts (a tripartite), to stay balanced, the same thing must be done to all three parts (rather than just both sides) to isolate the variable in the middle. The graph then is simply the values between the numbers with appropriate indication on the ends. Example 10: Graph and write interval notation for the conjunction: -9 < x < 1 Notice when graphing, a solid dot was used rather than the bracket. Unless indicated, either can be used. [-9, 1] Example 11: Solve, graph, and write interval notation: 2 > -4x + 2 > -6 -2 - 2 -2 Subtract 2 from all three parts 0 > -4x > -8 Divide all three parts by -4 -4 -4 -4 0< x < 2 Dividing by a negative flips the inequality signs (0, 2] Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)