WARM UP Graph the following inequalities: a. x ≤ 1 b. x < 2 3. – 4x – 3 < (1 – x ) ≥ 3

COMPOUND INEQUALITIES

OBJECTIVES  Solve and graph conjunctions of inequalities  Solve and graph disjunction of inequalities  Solve more complex inequality problems.

VOCABULARY  and  compound inequality  inequalities  conjunction  disjunction  empty set  intersection  or  union

COMPOUND INEQUALITIES  A statement such as the moon is red and the night is cold may be true depending upon whether the individual statements are true.  In mathematics and elsewhere, a compound statement may be composed of several shorter statements, each connected by the word and or the word or.  Similarly, the compound inequality x 0 may be true for a value of x, depending upon the truth of the individual statements.

CONJUNCTIONS  Here are some examples:  When two or more statements are joined by the word and, the new compound statement is called a conjunction. The moon is red and the night is cold x + y = 5 and x – y = 2 -2 < x and x < 1  For a conjunction to be true, all of the individual statements must be true.

EXAMPLE -2 < x and x < 1 Let’s look at the solutions sets of graphs of a conjunctions and their individual statements. For a number to be a solution of the conjunction, it must be in both solution sets. -2 < x and x < 1

EXAMPLE CONTINUED -2 < x and x < 1 The solution set of the conjunction contains the elements common to both of the individual solutions sets. The conjunction “-2 < x a nd x < 1” can be abbreviated -2 < x < 1. {x|-2 < x and x < 1}

INTERSECTIONS  For two sets A and B, we can represent the intersection by A B.  The set of common elements of two or more sets is called the intersection of the sets. {x|-2 < x and x < 1 =  The solution set of -2 < x and x < 1 is the intersection of the solution set.  If sets have no common members, the intersection is the empty set, which can be represented by the symbol.

EXAMPLE -2 < x and x < 1 Graph -3 ≤ x < 4 The graph of the intersection The conjunction corresponds to an intersection of sets. The solution is the intersection of the solution sets.

TRY THIS…. Grapha. -1 ≤ x -2 < y < 5

MORE EXAMPLES We write the conjunction with the word and : Solve:-3 < 2x + 5 < 7 -3 < 2x + 5 and 2x + 5 < 7 We could solve the individual inequalities separately and abbreviate the answer: -3 + (-5) < 2x (-5) and -8 < 2x 2x < 2 and -4 < x and x < 1 -4 < x < 1

EXAMPLE CONTINUED Note that we did the same thing to each inequality to every step. We can simplify the procedure as follows: -3 < 7< 2x (-5) < 2x (-5) < 7 + (-5) -8< 2x< 2 -4 < x < 1 The solution set is {x|-4 < x < 1}

TRY THIS…. Solve:1. -2 ≤ 3x + 4 ≤ ≥ -x + 3 > -6

DISJUNCTIONS  When two or more statements are joined by the word or to make a compound statement, the statements is called a disjunction.  Here are some examples of disjunction: It’s raining or the wind is blowing  For a disjunction to be true, at least one of the individual statements must be true. y is an even number or y is a prime number x -3

EXAMPLE -2 < x and x < 1 Let’s look at the solutions sets of graphs of a disjunction and its individual statements. If a number is in e ither or b oth of the solutions sets, it is in the solution set of the disjunction. The solution set of the disjunction is the set we get by joining the other two sets. x 3 or x > 3

UNIONS  The set obtained by joining two or more sets is called their union.  For the two sets A and B, we can name the union A B.  The solution set of x 3 is the union. or x > 3 =

EXAMPLE -2 < x and x < 1 Graph x ≤ 2 or x ≥ 5. The graph consists of the union of their individual parts.

TRY THIS…. a. x ≤ -2 or x > 4 b. x < -4 or x ≥ 6 Graph:

MORE EXAMPLES We solve the individual inequalities separately, but we continue writing the word or: Solve:-2 – 5 ≥ -2 or x – 3 > 2 -2 – ≤ x – > or -2x ≤ 3 x > 5 or x > 5 The solution set is {x|x or x > 5}

TRY THIS…. a. x – 4 < -3 or x – 4 ≥ 3 b. -2x + 4 ≤ -3 or x + 5 < 3 Solve: