Chapter 2

Inequalities and Proof

Section 2-1

Solving Inequalities in One Variable

Transitive Property - If a < b and b < c, then a < c

Addition Property - If a < b, then a + c < b + c

Properties of Order

1. If a < b and c is positive, then ac < bc

Multiplication Property

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

5x + 17 < 2 5(3-t) < 7 - t

Solve each inequality and graph its solution set

Section 2-2

Solving Combined Inequalities

Conjunction-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.

Disjunctions

Solve 2t + 7 13 or 5t – 4 < 6.

2t + 7 13 or 5t – 4 < 6

Now solve each inequality and graph the solution set for the disjunction.

Disjunctions

Solve y -1 or y 3

y -1 or y 3

Now solve each inequality and graph the solution set for the disjunction.

Section 2-3

Problem Solving Using Inequalities

Solving Word Problems Using Inequalities

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

Phrase Translation

Example:

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

SolutionFour consecutive integers –

n + (n + 1) + (n + 2) + (n + 3) 10 < 4n + 6 < 201 < n < 14/4Which integers work?

Section 2-4

Absolute Value in Open Sentences

Absolute ValueThe distance between a number x

and zero on a number line

If | x | = 1, then x = 1 or -1If | x | < 1, then -1 < x < 1If | x | > 1, then x < -1 or x > 1

Example - Equality Solve |3x - 2| = 8To solve, set up two equations

3x – 2 = – 8 3x – 2 = 8 3x = – 6 3x = 10 x = – 2 x = 10/3The solution is {-2, 10/3}

Example - Inequality Solve |3 – 2t| < 5Set up a compound inequality

– 5 < 3 – 2t < 5 – 8 < – 2t < 2 4 > t > – 1

The solution set is { t: – 1 < t < 4}

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

Section 2-7

Theorems about Order and Absolute Value