Our LessonOur Lesson

Inequalities with Inequalities with AddAdd and and SubtractSubtract

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1)Seven roses less than Kiran

2)Half as many Oranges as last year

3)One more than a number is 845

4)Eleven fewer hours than her sister

5)One-third the population of California

Warm Up

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Translate verbal phrases in to algebraic expressions

Lets review what we have learned in the last lesson

He weighs 180 pounds, which is 22 pounds 180 = 2z-22less than double her weight

Five more hits than the Martins h + 5

Three more than 8 times as many trees is 75 trees 8t – 3 = 75

Example:

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Addition Subtraction

Words used to express Addition/Subtraction

PlusSumMore than Increased by Total

MinusDifferenceLess thanLessDecreased by

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Words used to express Multiplication/Division

Multiplication Division

TimesProductMultipliedOf TwiceThrice

DividedQuotient

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USING A VERBAL MODEL

In Mathematics there is a difference between a phrase and a sentence. Phrases translate into

expressions; sentences translate into equations or inequalities

Expressions

Phrases

Equations or Inequalities

Sentences

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USING A VERBAL MODELUSING A VERBAL MODEL

Phrase Expression

=

Sentence Equation

Sentence Inequality

The sum of six and a number is twelve.

6 + x = 12

In this sentence, “is” says thatone quantity is equal to one another

In this sentence, the words “is less than” indicate an inequality

The sum of six and a number is less than twelve.

6 + x < 12 is less than twelve.

6 + x < 12

The sum of six and a number 6 + xPhrase

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Inequalities are used in many things like nutrition, budgeting, Personal financing, and academics.

15+8+4+2 + g < 68

Grams of fat in snacks

Grams of fat remaining

Total Grams

That is, 29 + g < 68

The symbol < indicates less than or equal to. It is used in this situation because the total number of grams of fat in daily diet should be no greater than 68. If this were an equation, we would subtract 29 from each side.

Lets get Started

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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

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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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Procedures that leave the inequality symbol unchanged

• Simplify both sides of the inequality by combining like terms and eliminating parenthesis

• Add or subtract the same expression on both sides of the inequality

• Multiply or divide both sides of the inequality by the same positive expression

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Rules to Solve Linear Inequalities

• Rule No : 1 - Addition• If a < b, then a + c < b + c • 2 < 5 • and c = 5• a + c < b + c • 2 + 5 < 5 + 5 • 7 < 10

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ADDING & SUBTRACTING INEQUALITY

Steps:-To solve the inequality means isolate a variable x

and 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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Solve the inequality x – 7 < 8

Step1: Add 7 on both sides

x – 7 + 7 < 8 + 7

Step2: Add the numbers

x < 15

Inverse of -7

is + 7

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Rules to Solve Linear Inequalities

Rule No : 2 - Subtraction If a < b, then a - c < b - c 4 < 7 c = 2 a - c < b - c 4 – 2 < 7 - 2

2 < 5

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Solve the inequality y + 6 > -11

Step1 : Subtract 6 on both sides y + 6 – 6 > -11 – 6

Step2 : Simplify the inequality y > -17

Inverse of + 6 is - 6

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Word problems in Inequalities

Phrases Translation

is 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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Example

Bill is at least 21 years old

Sample Sentences Equivalent Form

Bill age is 21

At most 5 students dropped the course

Students ≤ 5

The film was between 90 and 100 minutes long. 90 < t <

100

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Break Time

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1) t - 7 < 14

2) r – 5 > 27

3) k - 34 ≤ 10

4) v – 22 ≥ -11

5) y - 3 < -8

Your Turn

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6) x + 5 < 14 7) z + 9 > -7

8) 9) c + 19 ≥ -13

10) u + 13 ≥ 25

Your Turn

Boston is more than 200 miles away

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1) Lucy earned $400 and $550 in interest, the last 2 years. How much interest must she earn this year so that her average earnings over the three year period is more than $600?

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2) Chris has scored a 85 and a 89 on her first two math 23 tests.  What must she get on her third test to have at least a total score of 270. 

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3) Kim decides to take up golf.  Her golf club membership will cost $450 for the season she spends $300 on the gear and now she wants to buy golf shoes. Kim has decided not to spend more than $1000 on golf for the season. How much at most can she pay for the shoes.

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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 today

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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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Word problems in Inequalities

Phrases Translation

is 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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ADDING & SUBTRACTING INEQUALITY

Steps:-To solve the inequality means isolate a variable x

and 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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You did great in your lesson today !

Be sure to practice and keep up the good work