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

Inequalities  Properties – Similar to those of Equations from Chapter 1.  Comparison Property – Exactly only one of the following statements is true: a b.  Transitive Property – If a < b and b < c, then a < c.  Addition Property – If a < b, then a + c < b + c.  Multiplication Property - 1. If a < b and c is positive, then ac < bc.  2. If a bc.

Inequalities  Inequality – A sentence formed by placing an inequality symbol between two expressions.  Inequality Symbols – one of the following symbols:  We do not solve inequalities but transform them as they do not have set solutions.  There are five steps to inequalities.

Steps to Transform Inequalities  Step 1 – Simplify by Distributing  Step 2 – Simplify by Combining Like Terms  Step 3 – Add and/or Subtract the same value on both sides of the inequality to isolate the variable term.  Step 4 - Multiply and/or Divide the same positive value on both sides of the inequality to isolate the variable term.  Step 5 - Multiply and/or Divide the same negative value on both sides of the inequality to isolate the variable term and reverse the direction of the inequality.

Inequalities  Examples –

Combined Inequalities  Conjunction – A sentence formed by joining two inequalities with the word and. A conjunction is true when both parts of the sentence are true. If only one sentence is true the conjunction is false.  Example: X > b and x b and x < a or a < x < b  -2 x  To solve a conjunction you find the values of the variable for which both parts of the sentence are true.

Combined Inequalities  Disjunction – A sentence formed by joining two inequalities with the word or. A disjunction is true when at least one of the sentences is true.  Example: x < b or x = b or  x < 2 or x = 2 or  To solve a disjunction you find the values of the variable for which at least one of the sentences are true.

Combined Inequalities  Examples – Conjunctions –

Combined Inequalities  Examples – Disjunctions –

Word Problems  Follow the same steps as with equations. They are:  Step 1 – Read the Problem  Step 2 – Draw a picture/diagram/graph  Step 3 – Define your variable  Step 4 – Label picture/diagram/graph  Step 5 – Re-Read the problem  Step 6 – Set up Equation  Step 7 – Solve  Step 8 – Check your answer  Note about Step 7 – Please make sure you follow the transformation steps and graph the answer.

Word Problems  Phrases to know and their Translations PhraseTranslation X is at least a. X is no less than a. X is at most b. X is no greater than b. X is between a and b. X is between a and b inclusive. a < x < b

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

Word Problems  Define your Variable – x= number of passengers  Label your picture  Re-read problem  Set-up Equation  Solve – remember follow rules for inequalities and graph.  Check your answer.

Absolute Value and Inequalities  SentenceEquivalent Sentence Graph  |a| = 1a = -1 or a = 1 The distance between a and 0 is 1.  |b| > 1b 1 The distance between b and 0 is greater than 1.  |c| < 1-1 < c < 1 The distance between c and 0 is less than 1.

Absolute Value  Examples -

Absolute Value  Example – We can also just solve by graphing. -