A LGEBRA CHAPTER 3 Solving and Graphing Linear Inequalities

O NE - STEP LINEAR INEQUALITIES —3.1

V OCABULARY An equation is formed when an equal sign (=) is placed between two expressions creating a left and a right side of the equation An equation that contains one or more variables is called an open sentence When a variable in a single-variable equation is replaced by a number the resulting statement can be true or false If the statement is true, the number is a solution of an equation Substituting a number for a variable in an equation to see whether the resulting statement is true or false is called checking a possible solution

I NEQUALITIES Another type of open sentence is called an inequality. An inequality is formed when and inequality sign is placed between two expressions A solution to an inequality are numbers that produce a true statement when substituted for the variable in the inequality

I NEQUALITY S YMBOLS Listed below are the 4 inequality symbols and their meaning <Less than ≤Less than or equal to >Greater than ≥Greater than or equal to Note: We will be working with inequalities throughout this course…and you are expected to know the difference between equalities and inequalities

G RAPHS OF LINEAR INEQUALITIES Graph (1 variable) The set of points on a number line that represents all solutions of the inequality

G RAPHS OF LINEAR INEQUALITIES

W RITING LINEAR INEQUALITIES Bob hopes that his next math test grade will be higher than his current average. His first three test scores were 77, 83, and 86. Why would an inequality be best in this case? How can we come up with this inequality? Graph!

S OLVING ONE - STEP LINEAR INEQUALITIES Equivalent Inequalities Two or more inequalities with exactly the same solution Manipulating Inequalities All of the same rules apply to inequalities as equations * When multiplying or dividing by a negative number, we have to switch the inequality! Less than becomes greater than, etc.

S OLVING WITH ADDITION / SUBTRACTION

S OLVING WITH MULTIPLICATION / DIVISION

W HY DO WE HAVE TO CHANGE THE SIGN ? Is there another way we can solve this?

S OLVING MULTI - STEP LINEAR INEQUALITIES —3.2 A LGEBRA CHAPTER 3 Solving and Graphing Linear Inequalities

M ULTI STEP INEQUALITIES Treat inequalities just like you would normal, everyday equations* *change the sign when multiplying or dividing by a negative!!

E XAMPLES :

E XAMPLE You plan to publish an online newsletter that reports the results of snow cross competitions. You do not want your monthly costs to exceed $2370. Your fixed monthly costs are $1200. You must also pay $130 per month to each article writer. How many writers can you afford to hire in a month?

E XAMPLES : T RY THESE ON YOUR OWN !

o o ● ● Answer Now 1) W HICH GRAPH REPRESENTS THE CORRECT ANSWER TO > 1

2) W HEN SOLVING > -10 WILL THE INEQUALITY SWITCH ? 1. Yes! 2. No! 3. I still don’t know! Answer Now

3) W HEN SOLVING WILL THE INEQUALITY SWITCH ? 1. Yes! 2. No! 3. I still don’t know! Answer Now

4) S OLVE -8 P ≥ p ≥ p ≥ p ≤ p ≤ -12 Answer Now

o o ● ● Answer Now 5) S OLVE 7 V < -105

C LASS WORK : P.343 #15-37 ODD I F YOU DO NOT FINISH IN CLASS, THEN IT BECOMES HOMEWORK !

C OMPOUND INEQUALITIES —3.6 A LGEBRA CHAPTER 3 Solving and Graphing Linear Inequalities

C OMPOUND INEQUALITY What does compound mean? Compound fracture? So…what’s a compound inequality? An inequality consisting of two inequalities connected by an and or an or

G RAPHING C OMPOUND I NEQUALITIES Graph the following:

G RAPHING C OMPOUND I NEQUALITIES Graph the following:

G RAPHING C OMPOUND I NEQUALITIES Graph the following: All real numbers that are greater than or equal to -2 and less than 3

S OLVING C OMPOUND INEQUALITIES Again….treat these like equations! Whenever we do something to one side… …We do it to every side!

S OLVING C OMPOUND I NEQUALITIES

HOMEWORK : P.349 #12-36 EVEN

S OLVING A BSOLUTE -V ALUE E QUATIONS AND I NEQUALITIES —3.6 (D AY 1)

A BS. V ALUE What is Absolute Value? Distance from zero What does that mean?

A BS. V ALUE So….an absolute value equation has how many solutions? Is this always true?

A BS. V ALUE How do we apply this to equations? Ex:

E XAMPLES

P.356#19-36

S OLVING A BSOLUTE -V ALUE E QUATIONS AND I NEQUALITIES —3.6 (D AY 2)

A BSOLUTE V ALUE AND I NEQUALITIES

E XAMPLES