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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

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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

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Inequality Symbols

SYMBOL MEANING <     less than >    greater than

≤      less than or equal to

≥       greater than or equal to

 

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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

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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

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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

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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.

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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

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a (x)

a (5)?

If x < 5 is a (x) < a (5) ?

Multiplication

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(x) a

(5) a

?

If x < 5 is (x) < (5) a a

Division

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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

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Examples

8x < 296

8x ÷ 8 < 296 ÷ 8

x < 37

y 4 < 45

y 14 4 < 45 ÷÷

Y < 180

14

Use Division Use Multiplication

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When solving Inequalities, if you multiply and divide each side by the same negative number, you must reverse the direction of the inequality symbol.

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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

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x

5

If x < 5, what about the opposite of x and 5?

-x

-5

Opposite sign

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x

5

If x < 5, what about the opposite of x and 5?The answer:

-5

-x

- x > -5 It’s the opposite!

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So if you multiply or divide by a negative number:

REVERSE the symbol symbol !

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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

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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

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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

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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

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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

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Game time

Click here to play a game

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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 %

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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

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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.

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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.

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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

<

>

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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

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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

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You had a great lesson today !

Be sure to practice what you have learned