InequalitiesInequalities

Objective - TSWBAT Solve simple Objective - TSWBAT Solve simple inequalities in one variable and inequalities in one variable and conjunctions and disjunctions.conjunctions and disjunctions.

InequalitiesInequalities Properties – Similar to those of Equations from Properties – Similar to those of Equations from

Chapter 1.Chapter 1. Comparison Property – Comparison Property – Exactly only one of the Exactly only one of the

following statements is true: a < b, a = b, a > b.following statements is true: a < b, a = b, a > b. Transitive PropertyTransitive Property – If a < b and b < c, then a – If a < b and b < c, then a

< c.< c. Addition PropertyAddition Property – If a < b, then a – If a < b, then a

+ c < b + c.+ c < b + c. Multiplication PropertyMultiplication Property - 1. If a < b and c is - 1. If a < b and c is

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

InequalitiesInequalities

Inequality – A sentence formed by placing Inequality – A sentence formed by placing an inequality symbol between two an inequality symbol between two expressions.expressions.

Inequality Symbols – one of the following Inequality Symbols – one of the following symbols: symbols:

We do not solve inequalities but transform We do not solve inequalities but transform them as they do not have set solutions.them as they do not have set solutions.

There are five steps to inequalities.There are five steps to inequalities.

,,,,

Steps to Transform InequalitiesSteps to Transform Inequalities Step 1 – Simplify by DistributingStep 1 – Simplify by Distributing Step 2 – Simplify by Combining Like TermsStep 2 – Simplify by Combining Like Terms Step 3 – Add and/or Subtract the same value on Step 3 – Add and/or Subtract the same value on

both sides of the inequality to isolate the variable both sides of the inequality to isolate the variable term. term.

Step 4 - Multiply and/or Divide the same positive Step 4 - Multiply and/or Divide the same positive value on both sides of the inequality to isolate the value on both sides of the inequality to isolate the variable term. variable term.

Step 5 - Multiply and/or Divide the same negative Step 5 - Multiply and/or Divide the same negative value on both sides of the inequality to isolate the value on both sides of the inequality to isolate the variable term and reverse the direction of the variable term and reverse the direction of the inequality. inequality.

InequalitiesInequalities

Examples – Examples –

Combined InequalitiesCombined Inequalities

Conjunction – Conjunction – A sentence formed by A sentence formed by joining two inequalities with the word joining two inequalities with the word andand. . A conjunction is true when A conjunction is true when bothboth parts of parts of the sentence are true. If only one the sentence are true. If only one sentence is true the conjunction is false. sentence is true the conjunction is false.

Example: Example: X > bX > b and and x < a x < a oror a < x < b a < x < b -2 < x < 3-2 < x < 3 oror -2<x and 3>x -2<x and 3>x To solve a conjunction you find the values To solve a conjunction you find the values

of the variable for which of the variable for which bothboth parts of the parts of the sentence are true. sentence are true.

Combined InequalitiesCombined Inequalities

Disjunction –Disjunction – A sentence formed by A sentence formed by joining two inequalities with the word joining two inequalities with the word oror. A disjunction is true when at . A disjunction is true when at least least oneone of the sentences is true. of the sentences is true.

Example: Example: x < bx < b or or x = bx = b oror x < 2x < 2 or or x = 2x = 2 or or To solve a disjunction you find the To solve a disjunction you find the

values of the variable for which values of the variable for which at at least oneleast one of the sentences are true. of the sentences are true.

bx

2x

Combined InequalitiesCombined Inequalities

Examples – Conjunctions – Examples – Conjunctions –

Combined InequalitiesCombined Inequalities

Examples – Disjunctions – Examples – Disjunctions –

Word ProblemsWord Problems Follow the same steps as with equations. They Follow the same steps as with equations. They

are:are: Step 1 – Read the ProblemStep 1 – Read the Problem Step 2 – Draw a picture/diagram/graphStep 2 – Draw a picture/diagram/graph Step 3 – Define your variableStep 3 – Define your variable Step 4 – Label picture/diagram/graphStep 4 – Label picture/diagram/graph Step 5 – Re-Read the problemStep 5 – Re-Read the problem Step 6 – Set up EquationStep 6 – Set up Equation Step 7 – SolveStep 7 – Solve Step 8 – Check your answerStep 8 – Check your answer Note about Step 7 – Please make sure you follow Note about Step 7 – Please make sure you follow

the transformation steps and graph the answer.the transformation steps and graph the answer.

Word ProblemsWord Problems

Phrases to know and their Phrases to know and their TranslationsTranslations

PhrasePhrase TranslationTranslation

X is at least a.X is at least a.

X is no less than a. X is no less than a.

X is at most b.X is at most b.

X is no greater than b. X is no greater than b. X is between a and b.X is between a and b.

X is between a and b inclusive. X is between a and b inclusive. a < x < ba < x < b

ax

bx

bxa

Word ProblemsWord Problems

ExampleExample - A bus is to be chartered - A bus is to be chartered for the freshman class trip. The for the freshman class trip. The basic fare is $9.50 per passenger. If basic fare is $9.50 per passenger. If more than 20 people go, everyone’s more than 20 people go, everyone’s fare is reduced by $.30 for each fare is reduced by $.30 for each passenger over this number (20). At passenger over this number (20). At least how many people must go to least how many people must go to make the fare less than $7.50 per make the fare less than $7.50 per passenger?passenger?

Draw a picture/diagramDraw a picture/diagram

Word ProblemsWord Problems

Define your Variable – x= number of Define your Variable – x= number of passengerspassengers

Label your pictureLabel your picture Re-read problemRe-read problem Set-up EquationSet-up Equation Solve – remember follow rules for Solve – remember follow rules for

inequalities and graph.inequalities and graph. Check your answer.Check your answer.

Absolute Value and InequalitiesAbsolute Value and Inequalities SentenceSentence Equivalent SentenceEquivalent Sentence

GraphGraph |a| = 1|a| = 1 a = -1 or a = 1 a = -1 or a = 1

The distance between a and 0 is 1.The distance between a and 0 is 1.

|b| > 1|b| > 1 b < -1 or b > 1b < -1 or b > 1 The distance between b and 0 is greater than 1.The distance between b and 0 is greater than 1.

|c| < 1|c| < 1 -1 < c < 1-1 < c < 1 The distance between c and 0 is less than 1. The distance between c and 0 is less than 1.

Absolute ValueAbsolute Value

Examples - Examples -

Absolute ValueAbsolute Value

Example – We can also just solve by Example – We can also just solve by graphing. - graphing. -