Inequalities and Proof Chapter 2 Inequalities and Proof

Solving Inequalities in One Variable Section 2-1 Solving Inequalities in One Variable

Transitive Property - If a < b and b < c, then a < c Properties of Order Transitive Property - If a < b and b < c, then a < c Addition Property - If a < b, then a + c < b + c

1. If a < b and c is positive, then ac < bc Multiplication Property 1. If a < b and c is positive, then ac < bc 2. If a < b and c is negative, then ac > bc

Equivalent Inequalities Inequalities with the same solution set 2x + 5 < 13 and 2x < 8 and x < 4 4x > 2(3 + 2x) and 2x > 3 + 2x

Transformations that Produce Equivalent Inequalities Simplifying either side of an inequality.

Transformations that Produce Equivalent Inequalities Adding to (or subtracting from) each side of an inequality, the same number or the same expression.

Transformations that Produce Equivalent Inequalities Multiplying (or dividing) each side of an inequality by the same negative number and reversing the inequality.

Transformations that Produce Equivalent Inequalities Multiplying (or dividing) each side of an inequality by the same positive number

Examples Solve each inequality and graph its solution set 5x + 17 < 2 5(3-t) < 7 - t

Solving Combined Inequalities Section 2-2 Solving Combined Inequalities

Conjunction- Example: A sentence formed by joining two sentences with the word and. In a conjunction both sentences are true. Example: Graph the solution set of the conjunction x > -2 and x < 3

Disjunction- A sentence formed by joining two sentences with the word or. It is true when at least one of the sentences is true. Example: Graph the solution set for the disjunction x < 2 or x = 2

Conjunctions in a Different form Solve 3 < 2x + 5 ≤ 15. First rewrite the conjunction with and. 3 < 2x + 5 and 2x + 5 ≤ 15 Now solve each inequality and graph the solution set for the conjunction.

Conjunctions in a Different form Solve -3 < -2(t -3) < 6. First rewrite the conjunction with and. -3 < -2(t-3) and -2(t-3) < 6 Now solve each inequality and graph the solution set for the conjunction.

2t + 7  13 or 5t – 4 < 6 Disjunctions Solve 2t + 7  13 or 5t – 4 < 6. Now solve each inequality and graph the solution set for the disjunction. 2t + 7  13 or 5t – 4 < 6

y  -1 or y  3 Solve y  -1 or y  3 Disjunctions Now solve each inequality and graph the solution set for the disjunction. y  -1 or y  3

Problem Solving Using Inequalities Section 2-3 Problem Solving Using Inequalities

Solving Word Problems Using Inequalities Phrase Translation x is at least a x is no less than a x ≥ a x is at most b x is no greater than b x ≤ b x is between a and b x is between a and b, inclusive a < x < b a ≤ x ≤ b

Example: Find all sets of 4 consecutive integers whose sum is between 10 and 20.

Solution Four consecutive integers – n + (n + 1) + (n + 2) + (n + 3) Which integers work?

Absolute Value in Open Sentences Section 2-4 Absolute Value in Open Sentences

Absolute Value The distance between a number x and zero on a number line If | x | = 1, then x = 1 or -1 If | x | < 1, then -1 < x < 1 If | x | > 1, then x < -1 or x > 1

Example - Equality To solve, set up two equations Solve |3x - 2| = 8 3x – 2 = – 8 3x – 2 = 8 3x = – 6 3x = 10 x = – 2 x = 10/3 The solution is {-2, 10/3}

The solution set is { t: – 1 < t < 4} Example - Inequality Solve |3 – 2t| < 5 Set up a compound inequality – 5 < 3 – 2t < 5 – 8 < – 2t < 2 4 > t > – 1 The solution set is { t: – 1 < t < 4}

Solving Absolute Value Sentences Graphically Section 2-5 Solving Absolute Value Sentences Graphically

Facts The distance between x and 0 on a number line is | x | The distance between the graphs of real numbers a and b is | a – b |, or | b – a |

Examples Solve |5 - x| = 2 {3, 7}

Examples Solve |b + 5| > 3 {b: b < -8 or b > -2}

Examples Solve |2n + 5| ≤ 3 {n: n ≤ -4 or n ≥ -1}

Section 2-6 Theorems and Proofs

Definitions Theorem - A statement that can be proved Corollary – A theorem that can be proved easily from another Axioms – Statements that we assume to be true (these are also called postulates)

Cancellation Property of Addition For all real numbers a, b, and c: If a + c = b + c, then a = b If c + a = c + b, then a = b

Cancellation Property of Multiplication For all real numbers a and b, and nonzero real numbers c: If ac = bc, then a = b If ca = cb, then a = b

Zero – Product Property For all real numbers a and b: ab = 0 if and only if a = 0 or b = 0

Theorems about Order and Absolute Value Section 2-7 Theorems about Order and Absolute Value