© 2013 KMS Technology. AGILE DATA WAREHOUSE DESIGN Dao Vo Confidential2.
Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j...
-
Upload
angelica-woods -
Category
Documents
-
view
213 -
download
0
description
Transcript of Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j...
![Page 1: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/1.jpg)
![Page 2: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/2.jpg)
Confidential 2
1) 81 > b -90 b < 171 2) s -12 > 8 s > 20
3) n - 3 ≤ 8 n ≤ 11
4) j – 7 ≥ 5 j ≥ 12
5) f - 83 < - 56 f < 27
Warm up
![Page 3: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/3.jpg)
Confidential 3
An inequality is a statement with a symbol of < , ≤, > or ≥ between numerical or variable expressions.
Example: x + 7 ≤ 11. x is a variable. xx is a very common variable that is
used in algebra, but you can use any letter (a, b, c, d, ....) to be a variable
Defining Inequality
Lets review what we have learnt in the last lesson
![Page 4: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/4.jpg)
Confidential 4
Inequality Symbols
SYMBOL MEANING < less than > greater than
≤ less than or equal to
≥ greater than or equal to
![Page 5: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/5.jpg)
Confidential 5
Rule No : 1 - Addition If a < b, then a + c < b + c
Rules to Solve Linear Inequalities
Rule No : 2 - Subtraction If a < b, then a - c < b - c
![Page 6: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/6.jpg)
Confidential 6
ADDING & SUBTRACTING INEQUALITY
Steps:-
To solve the inequality means isolate a variable xand move the numbers to the other side of theinequality.
To move the number perform the reverseoperation. Addition is the reverse of Subtraction Subtraction is the reverse of Addition
![Page 7: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/7.jpg)
Confidential 7
Word problems in Word problems in InequalitiesInequalities
Phrases Translationis at least ≥is at most ≤
cannot exceed ≤must exceed >is between < and <
To solve the word problem translate the phrases and solve it
![Page 8: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/8.jpg)
Confidential 8
Just like with equations, the solution to an inequality is a value that makes the inequality true.
You can solve inequalities in the same way you can solve equations, by following these rules.
![Page 9: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/9.jpg)
Confidential 9
If c is positive and a < b, then ac < bc and a b , c = 0 c c
3. If c is positive and a > b, then ac > bc
and a b , c = 0 c c
<
>
For all numbers a, b and c, the following rules are true
![Page 10: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/10.jpg)
Confidential 10
a (x)
a (5)?
If x < 5 is a (x) < a (5) ?
Multiplication
![Page 11: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/11.jpg)
Confidential 11
(x) a
(5) a
?
If x < 5 is (x) < (5) a a
Division
![Page 12: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/12.jpg)
Confidential 12
Since this inequality involves multiplication, we must use the inverse, or division, to solve it.
We'll divide both sides by 16 in order to leave x alone on the left side.
16x > 512
1616 x > 512
16
x > 32
![Page 13: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/13.jpg)
Confidential 13
Examples
8x < 296
8x ÷ 8 < 296 ÷ 8
x < 37
y 4 < 45
y 14 4 < 45 ÷÷
Y < 180
14
Use Division Use Multiplication
![Page 14: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/14.jpg)
Confidential 14
When solving Inequalities, if you multiply and divide each side by the same negative number, you must reverse the direction of the inequality symbol.
![Page 15: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/15.jpg)
Confidential 15
If c is negative and a < b, then ac > bc
and a b c c
If c is negative and a > b, then ac < bc and a b c c
c = 0
c = 0
>
<
Opposite sign
![Page 16: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/16.jpg)
Confidential 16
x
5
If x < 5, what about the opposite of x and 5?
-x
-5
Opposite sign
![Page 17: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/17.jpg)
Confidential 17
x
5
If x < 5, what about the opposite of x and 5?The answer:
-5
-x
- x > -5 It’s the opposite!
![Page 18: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/18.jpg)
Confidential 18
So if you multiply or divide by a negative number:
REVERSE the symbol symbol !
![Page 19: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/19.jpg)
Confidential 19
If there is a negative number on one side of the inequality then we need to multiply/ divide by a negative number and reverse the symbol Since this inequality involves multiplication, we must use the inverse, or division, to solve it. We'll divide both sides by –16 in order to leave x alone on the left side. 16/16 x > 512/16
When we simplify, because we're dividing by a negative number, we have to remember to reverse the symbol. This gives "x is less than –32," not "x is greater than –32.“ x < -32
-16 x > 512
![Page 20: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/20.jpg)
Confidential 20
Example45 < 60
Perform the multiplication operation on each side of the equalityMultiply by 2
45 < 6045(2) < 60(2)
90 < 120True
Multiply by -2
45 < 6045(-2) < 60(-2)
-90 < -120False-90 > -120 True
![Page 21: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/21.jpg)
Confidential 21
Example45 < 60
Perform the division operation on each side of the equalityDivide by 5 Divide by -5
45 < 6045 60 5 5
<
9 < 12
45 < 6045 60-5 -5
<-9 < -12True False-9 > -12True
![Page 22: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/22.jpg)
Confidential 22
Your Turn !Solve each Inequality
1. 5c < 20 c < 4
2. 6m > 12 m > 2
3. 9j > 81 j > 9
4. Three times a number is more than 18 n > 6
5. $8 was less than 2/3 of the total m < 12
money spent
![Page 23: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/23.jpg)
Confidential 23
Your turn6.-11b ≥ -30 b ≤
30/11
7.y/-5 > 12 y < -60
8. -868 < -16r r < 217/4
9.f/17 ≥ 20.2 f ≥ 343.4
10. 34 > 9z z < 34/9
![Page 24: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/24.jpg)
Confidential 24
![Page 25: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/25.jpg)
Confidential 25
Game time
Click here to play a game
![Page 26: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/26.jpg)
Confidential 26
1. Alyssa's boss told her that if her department's income decreases more than eleven percent from last year's income she will not receive a bonus. Alyssa's department made $137,295 for the company last year. At the end of this year, the company accountant told Alyssa that her department's income was $124,938. Will Alyssa receive a bonus this year?
yes, she will receive a bonus as her department ‘s income decreased 9 %
![Page 27: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/27.jpg)
Confidential 27
2. Jeb wants to rent a car for his vacation. Value Cars rents cars for $25 per day plus $0.25 per mile. How far can he drive for one day if he wants to spend no more that $200 on car rental?
up to 700 miles
![Page 28: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/28.jpg)
Confidential 28
3. Alida has at most $10.50 to spend at a convenience store. She buys a bag of potato chips and a can of soda for $1.55. If gasoline at this store costs $1.35 per gallon, how many gallons of gasoline can Alida buy for her car, to the nearest tenth of a gallon?1.55 + 1.35g ≤ 10.50
g ≤ 6.6 Alida will have enough money if she gets no morethan 6.6 gallons of gasoline.
![Page 29: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/29.jpg)
Confidential 29
Just like with equations, the solution to an inequality is a value that makes the inequality true.
You can solve inequalities in the same way you can solve equations, by following these rules.
![Page 30: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/30.jpg)
Confidential 30
For all numbers a, b and c, the following rules are true
1. If c is positive and a < b, then ac < bc and a b , c = 0 c c
3. If c is positive and a > b, then ac > bc and a b , c = 0 c c
<
>
![Page 31: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/31.jpg)
Confidential 31
Since this inequality involves multiplication, we must use the inverse, or division, to solve it.
We'll divide both sides by 6 in order to leave x alone on the left side.
18 > 6k
18 6 > 6k
6
k < 3
![Page 32: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/32.jpg)
Confidential 32
When solving Inequalities, if you multiply and divide each side by the same negative number, you must reverse the direction of the inequality symbol.
When we simplify, because we're dividing by a negative number, h < -3
-4h > 12
-4h > 12 4 4
![Page 33: Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up.](https://reader033.fdocuments.net/reader033/viewer/2022051717/5a4d1b5f7f8b9ab0599ac922/html5/thumbnails/33.jpg)
Confidential 33
You had a great lesson today !
Be sure to practice what you have learned