Do Now 12/16/09Do Now 12/16/09

Take out HW from Monday and Tuesday.Take out HW from Monday and Tuesday. Text p. 366, #16-32 even, 36 & 40Text p. 366, #16-32 even, 36 & 40 Text p.368, #1-10Text p.368, #1-10

Copy HW in your planner.Copy HW in your planner. Text p. 372, #10-24 even, 30-34 even, 38, 40Text p. 372, #10-24 even, 30-34 even, 38, 40

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

16) d ≥ -37.216) d ≥ -37.2 18) k ≤ 0.4918) k ≤ 0.49 20) x ≥ 37.520) x ≥ 37.5 22) y ≤ 4522) y ≤ 45 24) p ≤ -0.3824) p ≤ -0.38 26) t < 0.6126) t < 0.61 28) The inequality sign was not 28) The inequality sign was not

reversed when dividing by -15. x < -3.reversed when dividing by -15. x < -3. 30) 8x > 50; x > 25/430) 8x > 50; x > 25/4 32) v/-9 < -18; v > 16232) v/-9 < -18; v > 162 36) at most 5 CDs36) at most 5 CDs 40) d < 18(0.75); d < 13.5 miles40) d < 18(0.75); d < 13.5 miles

Punchline worksheets Punchline worksheets ““How Can you Tell Van Winkle’s Trousers?”How Can you Tell Van Winkle’s Trousers?”

““How Did the Turtle Call for HELP When His Car Broke Down?” How Did the Turtle Call for HELP When His Car Broke Down?”

He Used His Shell PhoneHe Used His Shell Phone

There’s a rip in themThere’s a rip in them

HomeworkHomework Text p.368, #1-10 Text p.368, #1-10

1) x 1) x ≥ -13≥ -13 2) y < 82) y < 8 3) v ≥ -33) v ≥ -3 4) w > 134) w > 13 5) r < 85) r < 8 6) s > -316) s > -31 7) m ≤ -137) m ≤ -13 8) n < 288) n < 28 9) c ≤ -489) c ≤ -48 10) at least 50 minutes 10) at least 50 minutes

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

2y – 8 < 12 2y – 8 < 12

13 * 22 13 * 22 ≥≥ 14 - z14 - z

w/11 - 10 w/11 - 10 ≤≤ 121121

5c + 14 > 30

5c + 14 > 30

Section 6.3Section 6.3“Solve Multi-Step Inequalities”“Solve Multi-Step Inequalities”

Section 6.3Section 6.3“Solve Multi-Step Inequalities”“Solve Multi-Step Inequalities”

Solving Multi-Step Inequalities Solving Multi-Step Inequalities

STEP 1- Use distributive property and combine like terms.STEP 1- Use distributive property and combine like terms.STEP 2- Collect variables on one side of the inequality.STEP 2- Collect variables on one side of the inequality.STEP 3- “Undo” addition and/or subtraction.STEP 3- “Undo” addition and/or subtraction.STEP 4- “Undo” multiplication and/or division.STEP 4- “Undo” multiplication and/or division.STEP 5- Solve for the variable.STEP 5- Solve for the variable.STEP 6- Check your work.STEP 6- Check your work.

The steps for solving two-step and multi-step The steps for solving two-step and multi-step equations can be applied to linear inequalities.equations can be applied to linear inequalities.

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!!

REMEMBER!!!!!REMEMBER!!!!!

33xx – 7 – 7 << 8 8 Write original inequality.Write original inequality.

33xx << 15 15 Add Add 77 to each side. to each side.

xx << 5 5 Divide each side by Divide each side by 3.3.

ANSWERANSWER

The solutions are all real numbers less than The solutions are all real numbers less than 5.5. Check Check by substituting a number less than by substituting a number less than 55 in the original in the original inequality.inequality.

33xx – 7 – 7 << 8 8.. Graph your solutionGraph your solution..SolveSolve

Write original inequality.Write original inequality.––0.60.6((xx –– 5)5) << 1515 – –

––0.60.6xx + + 3 3 << 1515–– Distributive propertyDistributive property

Subtract Subtract 33 from each side. from each side.

–– x x >> ––2020 Divide each side by Divide each side by 0.6.0.6. Reverse Reverseinequality symbol.inequality symbol.

– –

–– 0.60.6xx << 1212––

SolveSolve –– 0.6(0.6(xx –– 5) 5) << 1515 – –

ANSWERANSWER

The solutions are all real numbers greater than equal The solutions are all real numbers greater than equal -20-20.. Check by substituting a number more than Check by substituting a number more than -20-20 in in the original inequality.the original inequality.

Write original inequality.Write original inequality.

Distributive property.Distributive property.

ANSWERANSWER

The solutions are all real numbers greater than The solutions are all real numbers greater than 20.20. Check by substituting a number greater than Check by substituting a number greater than 20.20.

––1144

( ( pp –12) > -2 –12) > -2

– – ¼ ¼ pp + 3 > -2 + 3 > -2

-¼ -¼ p p > –5 > –5

p p < 20< 20

––1144

( ( pp –12) > –2 –12) > –2

66xx – 7 – 7 >> 2 2x+x+1717 Write original inequality.Write original inequality.

66x x >> 2 2x+x+2424 Add Add 77 to each side. to each side.

44x x >> 24 24 Subtract Subtract 22xx from each side.from each side.

x x >> 6 6 Divide each side by Divide each side by 4.4.

ANSWER

The solutions are all real numbers greater than 6.

6x – 7 > 2x+17. Graph your solution.Solve

1414xx + 5 < 7(2+ 5 < 7(2x x –– 3)3) Write original inequality.Write original inequality.

1414xx + 5 < 14+ 5 < 14x x –– 2121 Distributive propertyDistributive property

5 < –5 < – 2121 Subtract Subtract 1414xx from each side.from each side.

There are no solutions because There are no solutions because 55 << –– 2121 is false.is false.

Solve: Solve: 1414xx + 5 < 7(2+ 5 < 7(2x x –– 3)3)

**HINT****HINT**If an inequality is equivalent to an inequality that is false, If an inequality is equivalent to an inequality that is false, such as 5 < -21, then the solution of the inequality has such as 5 < -21, then the solution of the inequality has NO SOLUTION. NO SOLUTION.

1212x x –– 1 > 6(21 > 6(2x x – 1)– 1) Write original inequality.Write original inequality.

Distributive propertyDistributive property

Subtract Subtract 1212xx from each side.from each side.

1212x x –– 1 > 121 > 12x x – 6– 6

–– 1 > –1 > – 66

All real numbers are solutions because All real numbers are solutions because –– 11 > > –– 66 is true. is true.

1212x x –– 1 > 6(21 > 6(2x x – 1)– 1)

**HINT****HINT**If an inequality is equivalent to an inequality that is true, If an inequality is equivalent to an inequality that is true, such as -1 > -6, then the solutions of the inequality aresuch as -1 > -6, then the solutions of the inequality areALL REAL NUMBERS . ALL REAL NUMBERS .

Graphs of “No Solution” and Graphs of “No Solution” and “All Real Numbers”“All Real Numbers”

““No Solution”No Solution”

““All Real Numbers”All Real Numbers”

Use the sign shown. A gas station charges $.10 less per gallon of gasoline if a customer also gets a car wash. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most $20?

Car Wash

STEP STEP 11

Write a verbal model. Then write an inequality.

Because you are getting a car wash, you will pay $2.09 – 2 $.10 = $1.99 per gallon of gasoline. Let g be the amount (in gallons) of gasoline that you buy.

1.99 +g 8 20<–

Price ofgasoline

(dollars/gallon)

Amount ofgasoline (gallons)

Price ofcar wash (dollars)

Maximumamount (dollars)

+ <•

STEPSTEP 22Solve the inequality.Solve the inequality.

Write inequality.Write inequality.

SubtractSubtract 88 from each side.from each side.1.991.99g g ≤ 12≤ 12

1.991.99g g ++ 8 ≤ 208 ≤ 20

Divide each side by Divide each side by 1.991.99..g g ≤ 6.03015.≤ 6.03015. . . . .

You can buy up to slightly more than You can buy up to slightly more than 66 gallons of gasoline. gallons of gasoline.

CHECKCHECK

You can use a table to check the You can use a table to check the reasonableness of your reasonableness of your answer.The table shows that you answer.The table shows that you will pay will pay $19.94$19.94 for exactly for exactly 6 6 gallons gallons of gasoline. Because of gasoline. Because $19.94$19.94 is less is less than than $20,$20, it is reasonable to it is reasonable to conclude that you can buy slightly conclude that you can buy slightly more than more than 66 gallons of gasoline. gallons of gasoline.

GasolineGasoline

(gal)(gal)

Total amount spentTotal amount spent

(dollars)(dollars)

00 8.008.00

11 9.999.99

22 11.9811.98

33 13.9713.97

44 15.9615.96

55 17.9517.95

66 19.9419.94

HomeworkHomework

Text p. 372, #10-24 even, 30-34 even, 38, Text p. 372, #10-24 even, 30-34 even, 38, 4040