Do Now 12/17/10Do Now 12/17/10

Copy HW in your planner.Copy HW in your planner. Text p. 359, #4-14 even 22-36 even.Text p. 359, #4-14 even 22-36 even. Text p. 366, #16-32 even, 36 & 40. Text p. 366, #16-32 even, 36 & 40.

In your notebook, answer the following question. In your notebook, answer the following question. Outside of the math classroom, where have you Outside of the math classroom, where have you heard phrases such as “at least” or “no more heard phrases such as “at least” or “no more than”? Give examples. How would you write than”? Give examples. How would you write the phrases mathematically?the phrases mathematically?

Chapter 6 Preview Chapter 6 Preview “Solving and Graphing Linear Inequalities”“Solving and Graphing Linear Inequalities”

(6.1) Solve Inequalities Using Addition and Subtraction(6.1) Solve Inequalities Using Addition and Subtraction

(6.2) Solve Inequalities Using Multiplication and Division(6.2) Solve Inequalities Using Multiplication and Division

(6.3) Solve Multi-Step Inequalities(6.3) Solve Multi-Step Inequalities

(6.4) Solve Compound Inequalities(6.4) Solve Compound Inequalities

(6.5) Solve Absolute Value Equations(6.5) Solve Absolute Value Equations

(6.6) Solve Absolute Value Inequalities(6.6) Solve Absolute Value Inequalities

(6.7) Graph Linear Inequalities in Two Variables(6.7) Graph Linear Inequalities in Two VariablesWinter break

INEQUALITIESINEQUALITIES – – mathematical sentence formed by mathematical sentence formed by placing a <, ≤, >, or ≥ between two placing a <, ≤, >, or ≥ between two expressions. expressions.

y² – 8 < 12 y² – 8 < 12

13 * 22 13 * 22 ≥≥ 14 - z14 - z

11 - a 11 - a ≤≤ 121121

5 ÷ x > 30

5 ÷ x > 30

Section 6.1Section 6.1“Solve Inequalities Using Addition and “Solve Inequalities Using Addition and

Subtraction”Subtraction”

Writing Equations with InequalitiesWriting Equations with Inequalities

SymbolSymbol MeaningMeaning Key phrasesKey phrases

== Is equal toIs equal to The same asThe same as

<< Is less thanIs less than Fewer thanFewer than

≤≤ Is less than or equal Is less than or equal to to

At most, no more At most, no more thanthan

>> Is greater thanIs greater than More thanMore than

≥≥ Is greater than or Is greater than or equal toequal to

At least, no less At least, no less thanthan

On a number line, the On a number line, the GRAPH OF AN GRAPH OF AN INEQUALITYINEQUALITY is the set of points that is the set of points that represent represent ALL SOLUTIONSALL SOLUTIONS of the of the inequality.inequality.

Graph x < 8Graph x < 8

55 66 77 88

Graph x ≥ 11Graph x ≥ 11

1212 13131111 14141010 1515

““Less than” and “greaterLess than” and “greaterthan” are representedthan” are representedwith an open circle.with an open circle.

““Less than or equal to” Less than or equal to” and “greater than or equaland “greater than or equalto” are represented with a to” are represented with a closed circle.closed circle.

101099 1111

88 99

SOLUTION

The highest temperature recorded in the United States The highest temperature recorded in the United States was was 134°F134°F at Death Valley, California, in at Death Valley, California, in 19131913. Use only . Use only this fact to write and graph an inequality that this fact to write and graph an inequality that describes the temperatures in the United States.describes the temperatures in the United States.

Let T represent a temperature (in degrees Fahrenheit) in the United States. The value of T must be less than or equal to 134. So, an inequality is T ≤ 134.

SOLUTION

Write an inequality represented by the graph.

The closed circle means that 8 is not a solution of the inequality. Because the arrow points to the left, all numbers less than 8 are solutions.

ANSWER

An inequality represented by the graph is x < 8.

SOLUTION

The closed circle means that – 2.5 is a solution of the inequality. Because the arrow points to the right, all numbers greater than – 2.5 are solutions.

ANSWER

An inequality represented by the graph is x > – 2.5.

Write an inequality represented by the graph.

Solving an Inequality…Solving an Inequality…

m + 4 < 12m + 4 < 12- 4 4 -44

m < 8m < 8

Isolate the variable! Get ‘m’ by itself.Isolate the variable! Get ‘m’ by itself.

To get the ‘m’To get the ‘m’by itself get rid ofby itself get rid of““adding 4.”adding 4.”

Do the opposite.Do the opposite.““Subtract 4.”Subtract 4.”

Whatever you do toWhatever you do toone side of the Inequalityone side of the Inequalityyou must do the other side.you must do the other side.

n - 5 ≥ 6n - 5 ≥ 6+ 5 + 5 +5 +5 n ≥ 11n ≥ 11

Isolate the variable! Get ‘n’ by itself.Isolate the variable! Get ‘n’ by itself.

To get the ‘n’To get the ‘n’by itself get rid ofby itself get rid of““subtracting 5.”subtracting 5.”

Do the opposite.Do the opposite.““Add 5.”Add 5.”

Whatever you do toWhatever you do toone side of the inequalityone side of the inequalityyou must do the other side.you must do the other side.

Solving an Inequality…Solving an Inequality…

x x – 5 > – 3.5– 5 > – 3.5+ 5+ 5 +5 +5

x x > 1.5> 1.5

ANSWERANSWER

The solutions are all real numbers greater The solutions are all real numbers greater than than 1.51.5. Check by substituting a number . Check by substituting a number greater than greater than 1.51.5 for for xx in the original inequality.in the original inequality.

Write original inequalityWrite original inequality..

Add Add 55 to each side to each side..

SimplifySimplify..

Solve x – 5 > -3.5 Solve x – 5 > -3.5 Graph your solutionGraph your solution

99 ≥ ≥ xx + 7+ 7 Write original inequalityWrite original inequality..

– – 77 –– 7 7 Subtract Subtract 77 from each side. from each side.

2 ≥ 2 ≥ xxSimplify.Simplify.

ANSWERANSWER

You can rewriteYou can rewrite 2 ≥ 2 ≥ x x asas x x ≥ ≥ 22. . The solutions are all real The solutions are all real numbers less than or equal numbers less than or equal to to 2.2.

Solve 9 Solve 9 ≥ x + 7≥ x + 7 Graph your solutionGraph your solution

Solve a real-world problemSolve a real-world problem

LUGGAGE WEIGHTSLUGGAGE WEIGHTS

You are checking a bag at an airportYou are checking a bag at an airport.. Bags can weigh Bags can weigh no more than no more than 5050 pounds pounds.. Your bag weighs Your bag weighs 16.816.8 pounds. pounds. Find the possible weightsFind the possible weights ww ((in poundsin pounds)) that you can that you can add to the bagadd to the bag..

SOLUTIONSOLUTION

Write a verbal modelWrite a verbal model.. Then write Then write and solve an inequalityand solve an inequality..

16.8 + 16.8 + w w ≤ ≤ 5050

16.8+ 16.8+ w w 5050<< Write inequality.Write inequality.

16.8 + 16.8 + w w –– 16.816.8 50 50 –– 16.816.8<< Subtract Subtract 16.816.8 from each side. from each side.

w w ≤ 33.2≤ 33.2 Simplify.Simplify.

ANSWERANSWER

You can add no You can add no more thanmore than 33.233.2 pounds.pounds.

Solve a real-world problemSolve a real-world problem

Write and solve an inequality to find Write and solve an inequality to find the possible values of x. the possible values of x.

Perimeter Perimeter ≤ 51.3 feet≤ 51.3 feet

14.2 ft14.2 ft

15.5 ft15.5 ft

x ftx ft

14.2 + 15.5 + x ≤ 51.3

29.7 + x ≤ 51.3

x ≤ 21.6

INEQUALITIESINEQUALITIES – – mathematical sentence formed by mathematical sentence formed by placing a <, ≤, >, or ≥ between two placing a <, ≤, >, or ≥ between two expressions. expressions.

y² – 8 < 12 y² – 8 < 12

13 * 22 13 * 22 ≥≥ 14 - z14 - z

11 - a 11 - a ≤≤ 121121

5 ÷ x > 30

5 ÷ x > 30

Section 6.2Section 6.2“Solve Inequalities Using Multiplication and “Solve Inequalities Using Multiplication and

Division”Division”

Write original inequalityWrite original inequality..

Divide each side by 7Divide each side by 7..

SimplifySimplify..

Solve . Graph your solutionSolve . Graph your solution917 x

7x7x > 91 > 917 7 7 7

x x > 13> 13

Graph x > 13Graph x > 13

1414 15151313 161612121010 1111

Write original inequalityWrite original inequality..

Multiply each side byMultiply each side by 4.4.

Simplify.Simplify.

ANSWERANSWER

The solutions are all real numbers The solutions are all real numbers less thanless than 2020.. Check by substituting Check by substituting a number less thana number less than 20 20 in the in the original inequality.original inequality.

Solve Graph your solution.Solve Graph your solution.

xx 44

<< 5.5.

xx 44

<<44 44 55

x x < 20< 20

54

x

Multiplying and/or dividing each Multiplying and/or dividing each side of an inequality by a NEGATIVE side of an inequality by a NEGATIVE number only produces an equivalent number only produces an equivalent inequality inequality IF the inequality sign is IF the inequality sign is REVERSED!!REVERSED!!

Solve Inequalities When Multiplying and Dividing Solve Inequalities When Multiplying and Dividing by a NEGATIVE”by a NEGATIVE”

Write original inequalityWrite original inequality..

Multiply each side byMultiply each side by –– 7. 7. Reverse inequality symbolReverse inequality symbol..

SimplifySimplify..

ANSWERANSWER

The solutions are all real numbers greater than The solutions are all real numbers greater than –– 11.2. 11.2. Check by substituting a number greater than Check by substituting a number greater than –– 11.2 11.2 in the in the original inequalityoriginal inequality..

m > –m > – 11.2 11.2

mm–– 77 < < 1.61.6

–– 77 1.61.6> > –– 7 7mm

–– 77

Solve . Graph your solution.Solve . Graph your solution.6.17

m

Write original inequalityWrite original inequality..

Divide each side by Divide each side by –3.–3. Reverse Reverseinequality symbolinequality symbol..

SimplifySimplify..

––33x x >> 2424..

––33xx––33

<< 2424––33

x x < – 8< – 8

Solve . Solve . 243 x

Solve a real-world problemSolve a real-world problem

SOLUTION

A student pilot plans to spend 80 hours on flight training to earn a private license. The student has saved $6000 for training. Which inequality can you use to find the possible hourly rates r that the student can afford to pay for training?

80r 6000 A < 80r 6000 B <– 80r 6000 C >– 80r 6000 D >

The total cost of training can be at most the amount of money that the student has saved. Write a verbal model for the situation. Then write an inequality.

The correct answer is B. A DCB

80 r <– 6000

Solve a real-world problemSolve a real-world problem

What are the possible hourly rates that the student can afford to pay for training?

Write inequality.Write inequality.

80r80

≤ 600080

Divide each side by Divide each side by 80.80.

r ≤ 75 Simplify.Simplify.

The student can afford to pay at most $75 per hour for training.

80 r ≤ 6000

Investigating Algebra ActivityInvestigating Algebra Activity“Inequalities with Negative Coefficients”“Inequalities with Negative Coefficients”

Complete the activity on page 362 in your Complete the activity on page 362 in your textbook. Read through the “Explore” textbook. Read through the “Explore” steps #1-3. Then complete the “Draw steps #1-3. Then complete the “Draw Conclusions” problems #1-8. Conclusions” problems #1-8.

HomeworkHomework

Text p. 359, #4-14 even 22-36 evenText p. 359, #4-14 even 22-36 even

2424

Text p. 366, #16-32 even, 36 & 40Text p. 366, #16-32 even, 36 & 40