Do Now 12/11/09Do Now 12/11/09

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.

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

x x – 5– 5 ++ 55 > – 3.5> – 3.5 ++ 55

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 in the original inequality.inequality.

Write original inequalityWrite original inequality..

Add Add 55 to each side to each side..

SimplifySimplify..

x x – 5 > – 3.5.– 5 > – 3.5. Graph your solutionGraph your solution..SolveSolve

SOLUTIONSOLUTION

p p – 9.2 < – 5– 9.2 < – 5

p –p – 9.2 9.2 ++ 9.2 9.2 < –< – 5 5 ++ 9.29.2

x <x < 4.2 4.2

Write original inequalityWrite original inequality..

Add Add 9.29.2 to each side to each side..

SimplifySimplify..

ANSWERANSWER The solutions are all real numbers less than The solutions are all real numbers less than 4.2.4.2. Check by substituting a number less Check by substituting a number less than than 4.2 4.2 for for xx in the original inequalityin the original inequality..

p p – 9.2 < – 5.– 9.2 < – 5. Graph your solutionGraph your solution..SolveSolve

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

99 –– 7 7 ≥≥ xx + 7 + 7 –– 7 7 Subtract Subtract 77 from each side. from each side.

2 ≥ 2 ≥ xx Simplify.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 Solve 99 xx + 7.+ 7. Graph your solution. Graph your solution.>>

Write original inequalityWrite original inequality..

Subtract Subtract 5.55.5 from each side from each side..

yy >> 0.5 0.5 SimplifySimplify..

yy ++ 5.5 5.5 >> 6 6

yy ++ 5.5 5.5 –5.5–5.5 >> 6 6 –5.5–5.5

ANSWERANSWER You can rewrite You can rewrite 0.5 < 0.5 < yy as as y >y > 0.5 . 0.5 . The The solutions are all real numbers greater than solutions are all real numbers greater than or equal to or equal to 0.5 .0.5 .

SOLUTIONSOLUTION

Solve Solve yy ++ 5.5 5.5 >> 6. 6. Graph your solution. Graph 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

Algebra in “Real-Life!”Algebra in “Real-Life!”

In your groups, complete the “real-life” In your groups, complete the “real-life” activity. activity. Interdisciplinary Application “Bald Eagles”Interdisciplinary Application “Bald Eagles”

Algebra in “Real-Life!”Algebra in “Real-Life!”

In your groups, complete the two “real-life” In your groups, complete the two “real-life” activities. activities. Interdisciplinary Application “Bald Eagles”Interdisciplinary Application “Bald Eagles”

• 1) y = 393x + 14,3241) y = 393x + 14,324• 2) graph2) graph• 3) approximately 18,2543) approximately 18,254• 4) habitat loss, less people counting 4) habitat loss, less people counting • 5) y = 512.5x + 13,6985) y = 512.5x + 13,698• 6) graph6) graph• 7) approximately 18,8807) approximately 18,880

HomeworkHomework

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

2424

NJASK7 PrepNJASK7 Prep

Practice Worksheet Form BPractice Worksheet Form B#2-24 even#2-24 even

2) x > -22) x > -2 4) x ≥ -74) x ≥ -7 6) x ≤ -16) x ≤ -1 8) n ≤ 128) n ≤ 12 10) m < 110) m < 1 12) x ≤ -7.512) x ≤ -7.5 14) y < 4.814) y < 4.8 16) d > 10.216) d > 10.2 18) m – 3 ≥ 10; m ≥ 1318) m – 3 ≥ 10; m ≥ 13 20) 85 > x – 63; x > 14820) 85 > x – 63; x > 148 22) 17 hats or less22) 17 hats or less 24) a). 6532 + 4887 < 5034 + s; s > 638524) a). 6532 + 4887 < 5034 + s; s > 6385

b). Yes, because 6392 > 6385. No, b). Yes, because 6392 > 6385. No, because 6385 is not > 6385. No, because 6377 because 6385 is not > 6385. No, because 6377 is not > 6385.is not > 6385.