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Rational Inequities Lesson 6.2.6.

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1 Rational Inequities Lesson 6.2.6

2 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. Let’s look… 3π‘₯+4=19 3π‘₯+4<19 βˆ’ 4 βˆ’ βˆ’ βˆ’ 4 3π‘₯= π‘₯< 15 π‘₯ = π‘₯ < 5 3π‘₯ 3 = 15 3 3π‘₯ 3 < 15 3

3 Question… Who can tell me 2 things that you do with inequalities that you don’t do with equations? 1) When multiplying or dividing both sides of the inequality by a negative number… you have to reverse (FLIP) the inequality symbol. 2) You graph inequalities.

4 Real-World Inequalities are used to describe a set of numbers.
In other words… you have more than one answer. There are many ways that inequalities are used in the Real-World. Let’s look at a some…

5 π‘π‘Žπ‘ π‘ π‘Žπ‘›π‘”π‘’π‘Ÿπ‘  ≀ 50 Bailong Elevator of China
This elevator runs along a cliff face in one of China’s National Forest Parks. It hold up to 50 passengers. π‘π‘Žπ‘ π‘ π‘Žπ‘›π‘”π‘’π‘Ÿπ‘  ≀ 50 Explain that even though the symbol is ≀ a negative number would not make sense in this situation.

6 Accommodates up to 4000 passengers. π‘Šβ„Žπ‘Žπ‘‘ π‘Ÿπ‘Žπ‘›π‘”π‘’ π‘œπ‘“ π‘›π‘’π‘šπ‘π‘’π‘Ÿπ‘  π‘Žπ‘Ÿπ‘’ 𝑀𝑒
0 to 4000 because not such thing as a negative person. Accommodates up to 4000 passengers. π‘Šβ„Žπ‘Žπ‘‘ π‘Ÿπ‘Žπ‘›π‘”π‘’ π‘œπ‘“ π‘›π‘’π‘šπ‘π‘’π‘Ÿπ‘  π‘Žπ‘Ÿπ‘’ 𝑀𝑒 π‘‘π‘’π‘Žπ‘™π‘–π‘›π‘” π‘€π‘–π‘‘β„Ž β„Žπ‘’π‘Ÿπ‘’?

7 𝑆𝑃𝐸𝐸𝐷 ≀12 1 2

8 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 Lavillenie’s world record, another athlete must exceed 6.16 meters. π‘š>6.16

9 Today… Today we are going to continue solving two-step inequalities, but today they will include Rational Numbers (decimals and fractions)

10 Let’s Get Started… Section 1: Quick Review of Two-Step Inequalities Section 2: Inequalities with Fractions Section 3: Inequalities with Decimals.

11 Solve and Graph 4x + 1 > 13 – 1 – 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, 100…(anything to the right of 3 on the number line) x > 3 Questions… 1) Is 3 a solution to this inequality? 2) Name some possible values for x.

12 If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed. Remember! –9π‘₯ + 7 ο‚³ 25

13 Solve and Graph –9x + 7 ο‚³ 25 – 7 – 7 Subtract 7 from both sides.
ο‚£ 18 Divide each side by –9; change ο‚³ to ο‚£. x ο‚£ –2 Questions… 1) Is -2 a solution to this inequality? 2) Name some possible values for x.

14 Moving On… Section 1: Quick Review of Two-Step Inequalities Section 2: Inequalities with Fractions Section 3: Inequalities with Decimals.

15 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. Let’s look at an example…

16 Solve and Graph + 2x 5 3 4 9 10 ο‚³ 20( + ) ο‚³ 20( ) 2x 5 3 4 9 10
20( + ) ο‚³ 20( ) 2x 5 3 4 9 10 Multiply both sides by the LCD, 20. 20( ) + 20( ) ο‚³ 20( ) 2x 5 3 4 9 10 Point out that the numbers β€œteamed-up” with x are positive, so don’t need to flip. Illustrate how to simplify so that we end up with an equation without fractions. Distributive Property. 8x + 15 ο‚³ 18 Since 15 is added to 8x, subtract 15 from both sides. – 15 – 15 8x ο‚³ 3

17 Since x multiplied by 8, divide both sides by 8.
Continued… 8x ο‚³ 3 ο‚³ 8x 8 3 Since x multiplied by 8, divide both sides by 8. x ο‚³ 3 8 3 8 Remind students how to plot a fraction on a number line.

18 Guided Practice #1 βˆ’ 2 3 π‘₯ ≀ 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‰₯ βˆ’10 3 4 what integers go here?

19 You Try #1 1 2 βˆ’ 𝑦 > 3 4 Solve in front of students. Write -1 and -2 in the boxes and then plot and graph the solution: 𝑦<βˆ’ 3 10

20 Guided Practice #2 3 8 π‘₯ βˆ’ 2 3 > 7 12
3 8 π‘₯ βˆ’ 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‘₯> π‘œπ‘Ÿ 9 1 3

21 You Try #2 3 4 π‘₯ βˆ’ ≀ βˆ’ 1 4 -1 and -2 go in the boxes. Then plot and graph the solution: π’™β‰€βˆ’ 𝟏 πŸ”

22 Guided Practice #3 6 7 < 1 7 π‘Ž + 53 56
6 7 < 1 7 π‘Ž 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 >βˆ’ 5 8

23 You Try #3 4 5 β‰₯ β„Ž 1 and 2 go in the boxes. Plot and graph the solution: β„Žβ‰€ 1 5

24 Last Section… Section 1: Quick Review of Two-Step Inequalities Section 2: Inequalities with Fractions Section 3: Inequalities with Decimals.

25 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. Let’s look at an example…

26 Solve and Graph 𝟎.πŸŽπŸ–π’Ž+πŸ“.πŸπŸ’ >𝟐.πŸ‘ 100(0.08π‘š + 5.14) > 100(2.3)
Multiply both sides by the 100. 100(0.08π‘š) +100(5.14) > 100(2.3) Distributive Property. 8π‘š > 230 Since 514 is added to 8m, subtract 514 from both sides. – – 514 8π‘š > βˆ’284

27 Since x multiplied by 8, divide both sides by 8.
Continued… 8m > -284 ο‚³ 8x 8 -284 Since x multiplied by 8, divide both sides by 8. x ο‚³ βˆ’πŸ‘πŸ“.πŸ“ Remind students how to plot a fraction on a number line.

28 Guided Practice #4 10 βˆ’6.4𝑧 < 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

29 You Try #4 𝑦 β‰₯ 23.16 2 and 3 go in the boxes. Plot and graph the solution: 𝑦β‰₯2.2

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