Do Now 1/3/11Do Now 1/3/11

Copy HW in your planner.Copy HW in your planner.– Text p. 141, #12-32 evensText p. 141, #12-32 evens

Be ready to copy POTW #6.Be ready to copy POTW #6.

ObjectiveObjective

SWBAT solve simple inequalities SWBAT solve simple inequalities and use inequalities to solve real-and use inequalities to solve real-life problemslife problems

In your notebook, answer the following In your notebook, 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?

Section 3.4, “ Solving Section 3.4, “ Solving Inequalities Using Addition and Inequalities Using Addition and

Subtraction”Subtraction”

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

y – 8 < 12 y – 8 < 12

13 13 ≥≥ 14 - z14 - z

11 + a 11 + a ≤≤ 121121

x +8 > 30

x +8 > 30

InequalitiesInequalities

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 GRAPH OF AN INEQUALITYAN INEQUALITY is the set of points is the set of points that represent that represent ALL SOLUTIONSALL SOLUTIONS of of the inequality.the inequality.

Graph x < 8Graph x < 8

55 66 77 88

Graph x ≥ 11Graph x ≥ 11

1212 13131111 14141010

““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

SOLUTIONSOLUTION

Write an inequality represented by the graph.Write an inequality represented by the graph.

The closed circle means that The closed circle means that 88 is not a solution of is not a solution of the inequalitythe inequality.. Because the arrow points to the Because the arrow points to the leftleft,, all numbers less than all numbers less than 88 are solutions are solutions..

ANSWERANSWER

An inequality represented by the graph is An inequality represented by the graph is x x << 88..

SOLUTIONSOLUTION

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

ANSWERANSWER

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

Write an inequality represented by the graph.Write an inequality represented by the graph.

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.

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

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

p <p < 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..

Solve p – 9.2 < -5 Solve p – 9.2 < -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

Write original inequalityWrite original inequality..

Subtract Subtract 5.55.5 from each side from each side..yy >> 0.5 0.5SimplifySimplify..

yy ++ 5.5 5.5 >> 6 6 – –5.5 –5.5 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 .

Solve y + 5.5 > 6 Solve y + 5.5 > 6 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

Write inequality.Write inequality.

Subtract Subtract 16.816.8 from each side. from each side.– – 16.8 –16.816.8 –16.8

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

w w ≤ 33.2≤ 33.2

16.8 + w 16.8 + w ≤ 50≤ 50

HomeworkHomework

Text p. 141, #12-38 evenText p. 141, #12-38 even