chapter 2 inequalities and proof section 2-1 solving inequalities in one variable
TRANSCRIPT
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