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Rational Inequities Lesson 6.2.6
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Prior⦠We have been learning about inequalities in Chapter 6.
You use the same steps when solving an inequality as you do when solving an equation. Letβs lookβ¦ 3π₯+4=19 3π₯+4<19 β 4 β β β 4 3π₯= π₯< 15 π₯ = π₯ < 5 3π₯ 3 = 15 3 3π₯ 3 < 15 3
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Questionβ¦ Who can tell me 2 things that you do with inequalities that you donβt do with equations? 1) When multiplying or dividing both sides of the inequality by a negative numberβ¦ you have to reverse (FLIP) the inequality symbol. 2) You graph inequalities.
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Real-World Inequalities are used to describe a set of numbers.
In other wordsβ¦ you have more than one answer. There are many ways that inequalities are used in the Real-World. Letβs look at a someβ¦
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πππ π ππππππ β€ 50 Bailong Elevator of China
This elevator runs along a cliff face in one of Chinaβs National Forest Parks. It hold up to 50 passengers. πππ π ππππππ β€ 50 Explain that even though the symbol is β€ a negative number would not make sense in this situation.
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Accommodates up to 4000 passengers. πβππ‘ πππππ ππ ππ’πππππ πππ π€π
0 to 4000 because not such thing as a negative person. Accommodates up to 4000 passengers. πβππ‘ πππππ ππ ππ’πππππ πππ π€π πππππππ π€ππ‘β βπππ?
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πππΈπΈπ· β€12 1 2
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Renaud Lavillenie is a French pole vaulter
Renaud Lavillenie is a French pole vaulter. He is the current world record holder, with a height of 6.16 m set February 15, Lavillenie won the gold medal at the 2012 Olympics. To break Lavillenieβs world record, another athlete must exceed 6.16 meters. π>6.16
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Today⦠Today we are going to continue solving two-step inequalities, but today they will include Rational Numbers (decimals and fractions)
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Letβs Get Startedβ¦ Section 1: Quick Review of Two-Step Inequalities Section 2: Inequalities with Fractions Section 3: Inequalities with Decimals.
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Solve and Graph 4x + 1 > 13 β 1 β 1 Since 1 is added to 4x, subtract 1 from both sides. 4x > 12 Since x is multiplied by 4, divide both sides by 4. 4x 4 > 12 No because open dot. 3.5, 4, 5, 100β¦(anything to the right of 3 on the number line) x > 3 Questionsβ¦ 1) Is 3 a solution to this inequality? 2) Name some possible values for x.
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If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed. Remember! β9π₯ + 7 ο³ 25
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Solve and Graph β9x + 7 ο³ 25 β 7 β 7 Subtract 7 from both sides.
ο£ 18 Divide each side by β9; change ο³ to ο£. x ο£ β2 Questionsβ¦ 1) Is -2 a solution to this inequality? 2) Name some possible values for x.
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Moving On⦠Section 1: Quick Review of Two-Step Inequalities Section 2: Inequalities with Fractions Section 3: Inequalities with Decimals.
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Clearing Fractions Reminder of steps⦠Step 1
Look at all of the denominators and find the LCD. Step 2 Multiply both sides of the inequality by the LCD. Letβs look at an exampleβ¦
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Solve and Graph + 2x 5 3 4 9 10 ο³ 20( + ) ο³ 20( ) 2x 5 3 4 9 10
20( + ) ο³ 20( ) 2x 5 3 4 9 10 Multiply both sides by the LCD, 20. 20( ) + 20( ) ο³ 20( ) 2x 5 3 4 9 10 Point out that the numbers βteamed-upβ with x are positive, so donβt need to flip. Illustrate how to simplify so that we end up with an equation without fractions. Distributive Property. 8x + 15 ο³ 18 Since 15 is added to 8x, subtract 15 from both sides. β 15 β 15 8x ο³ 3
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Since x multiplied by 8, divide both sides by 8.
Continuedβ¦ 8x ο³ 3 ο³ 8x 8 3 Since x multiplied by 8, divide both sides by 8. x ο³ 3 8 3 8 Remind students how to plot a fraction on a number line.
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Guided Practice #1 β 2 3 π₯ β€ 8 Do we need to Flip? Why or Why not? Solve in front on the students. Make sure they following along in their notes. Write the whole numbers -10 and -11 in the boxes with the pink lines, then illustrate how to plot and graph the answer: π₯ β₯ β10 3 4 what integers go here?
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You Try #1 1 2 β π¦ > 3 4 Solve in front of students. Write -1 and -2 in the boxes and then plot and graph the solution: π¦<β 3 10
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Guided Practice #2 3 8 π₯ β 2 3 > 7 12
3 8 π₯ β 2 3 > 7 12 Do we need to Flip? Why or Why not? Solve in front of students. Then fill in the boxes with 9 and 10 and then plot and graph the solution: π₯> ππ 9 1 3
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You Try #2 3 4 π₯ β β€ β 1 4 -1 and -2 go in the boxes. Then plot and graph the solution: πβ€β π π
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Guided Practice #3 6 7 < 1 7 π + 53 56
6 7 < 1 7 π Do we need to Flip? Why or Why not? Solve in front of students. Then fill in the boxes with -1 and -2. Then plot and graph the solution: a >β 5 8
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You Try #3 4 5 β₯ β 1 and 2 go in the boxes. Plot and graph the solution: ββ€ 1 5
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Last Section⦠Section 1: Quick Review of Two-Step Inequalities Section 2: Inequalities with Fractions Section 3: Inequalities with Decimals.
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Clearing Decimals Reminder of steps⦠Step 1
Find the decimal with the most digits. Step 2 Multiply both sides of the inequality by that power of 10. Skip this screen if you do not have you students clear the decimals. Letβs look at an exampleβ¦
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Solve and Graph π.πππ+π.ππ >π.π 100(0.08π + 5.14) > 100(2.3)
Multiply both sides by the 100. 100(0.08π) +100(5.14) > 100(2.3) Distributive Property. 8π > 230 Since 514 is added to 8m, subtract 514 from both sides. β β 514 8π > β284
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Since x multiplied by 8, divide both sides by 8.
Continuedβ¦ 8m > -284 ο³ 8x 8 -284 Since x multiplied by 8, divide both sides by 8. x ο³ βππ.π Remind students how to plot a fraction on a number line.
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Guided Practice #4 10 β6.4π§ < 2 Do we need to Flip? Why or Why not?
Solve in front of students. Then fill in the boxes with 1 and 2. Then plot and graph the solution: z >1.25
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You Try #4 π¦ β₯ 23.16 2 and 3 go in the boxes. Plot and graph the solution: π¦β₯2.2
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