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Unit 3 Seminar

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MM150 Unit 3 Seminar Agenda

• Sections 3.1 -3.4

Examples

• Variables: x, y, z, a• Algebraic Expression:

a + b

4x – 7

6y

x/4

They can be longer, like these:

3x2 – 7y3 + 12z – 2

a + b + c + d + e + f + g

4

Equations

• 2 + x = 11• 3y - 9 = 36• x/t = 64

• The solution to 2 + x = 11 is 9. We can check the solution by substituting 9 for x.

• 2 + x = 11• 2 + 9 = 11• 11 = 11 This is a true statement. 4

5

Equations

• What happens if we end up with a false statement?

• Is 10 a solution to 3y - 9 = 36? Check the solution.

• 3y - 9 = 36• 3(10) - 9 = 36• 30 - 9 = 36• 21 = 36 This statement is false. 5

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Evaluating Expressions• Exponents: • x2 AND 34 AND -7y3 AND 59

• 2*2*2*2*2*2*2, you can rewrite this as 27

x*x*x*x is x4

(2a)(2a)(2a) is (2a)3

(x + 6)(x + 6) is (x + 6)2

• x^2 is the same as x2

• 2^3 = 23 = 2*2*2 = 8

Be careful!(-2)4 = (-2)(-2)(-2)(-2) = 16

-24 = -(2*2*2*2) = -166

• Perimeter is the distance around a closed figure. The perimeter of a triangle can be written as a + b + c, where a, b, and c are the side lengths of the triangle.

Example: The sides of a triangle have lengths of 3 meters, 7 meters, and x meters. Determine the perimeter of the triangle if x is 10 meters .

Evaluate with x = 10

3 + 7 + 10 = 20 meters The perimeter of the triangle is 20 meters.

• Area is the measurement of surface measured in square units. The area of a rectangle can be written as l * w, where l is the length and w is the width.

Example: Find the area of a rectangular yard enclosed by a fence 12 yards long and 8 yards wide.

Evaluate with l = 12 and w = 8

12 * 8 = 96 square yards Therefore, the area is 96 square yards.

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EVERYONE:• Volume is space within a figure measured in cubed units. The volume of a cube can

be written as l * w * h, where l is the length, w is the width and h is the height.

Example: Find the volume of a cube with a length of 10 feet, a width of 4 feet and a height of 3 feet.

Evaluate with l = 10, w = 4 and h = 3

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EVERYONE: Answer• Volume is space within a figure measured in cubed units. The volume of a cube can

be written as l * w * h, where l is the length, w is the width and h is the height.

Example: Find the volume of a cube with a length of 10 feet, a width of 4 feet and a height of 3 feet.

Evaluate with l = 10, w = 4 and h = 3

10 * 4 * 3 = 120 cubic feet The volume is 120 cubic feet.

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Terms• Examples of terms:Constants: 3, -5, 0, 1/7, PiVariables: a, b, c, x, y, zProducts: 3x, ab2, -99ay5

Expressions can be one term (monomial): x, 5t, -10y

Expressions can have two terms (binomial): y + 9, -6s - 11

Expressions can have three terms (trinomial): x2 + 7x - 10

Expressions can have four terms or more (polynomial): x2y + xy - 11y + 23

NOTE: Decreasing power of the variable. 10

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Like and Unlike Terms

• 5x and 3x are like terms6ab and -9ab are like terms16x2 and x2 are like terms-0.35ac5 and -400ac5 are like terms

You can simplify like terms! For example,12a + 4a = 16a57x – 33x = 24x9x2 + 3x2 + x2 = 13x2

-ab + (-4ab) = -5ab

You cannot simplify unlike terms!!2x + 2y + 3x = 5x + 2y

8x2 – 4x + x2 = 9x2 – 4x 11

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Addition Property of Equality

For real numbers a, b, and c,

if a = b, then a + c = b + c.

Example: Non example:

If x = 4, If y = 9, then y + 7 = 9

then x + 2 = 4 + 2 Here we only added 7 to one side

Here we added 2 to both sides

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Solving Equations

• x - 7 = 18• x - 7 + 7 = 18 + 7• x = 25

• 12 = -4 + x• 12 + 4 = -4 + x + 4• 16 = x

• EVERYONE: 6 = x - 22. What is x? 13

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Example: 5 + 6 + x = 11 – 2

11 + x = 9

11 + x – 11 = 9 – 11

x = -2

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15

EVERYONE: solve for x: 2 – 8 = x – 5 – 1

-6 = x – 6

0 = x

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Multiplication Property of Equality

For real numbers a, b, and c, where c is not 0, if a = b, then a * c = b * c.

Example: Non example:

If x = 4, If y = 9, then y * 7 = 9

then x * 2 = 4 * 2 Here we only multiplied 7 to one side

Here we multiplied by 2 to both sides

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Solving Equations

• Example: (2/3)x = 4/5(3/2)(2/3)x = (3/2)(4/5)x = 12/10 x = 6/5

Example: x/6 = -1/26(x/6) = 6(-1/2)x = -6/2x = -3

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Division Property of Equality

For real numbers a, b, and c, where c is not 0, if a = b, then a/c = b/c.

Example: Non example:

If x = 4, If y = 9, then y/7 = 9

then x/2 = 4/2 Here we only divided one side

Here we divided both sides by 2

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Solving Equations

• Example: -3x = 18-3x/(-3) = 18/(-3)x = -6

Example: 9x = -89x/9 = -8/9x = -8/9

Example: -x = -3-1(-x) = -1(-3) -OR- -x/(-1) = -3/(-1)x = 3 x = 3

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20

3 = 15x + 20

-17 = 15x

-17/15 = x

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Example: 3 – 12x = 3x + 20

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EVERYONE: solve for x:22 + 3 – 6x = 2x + x + 11

25 – 6x = 3x + 11

25 = 9x + 11

14 = 9x

14/9 = x

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Example: 3(2x – 5) – 7 = x(x + 4) – x2

6x – 15 – 7 = x2 + 4x – x2

-22 = -2x

11 = x

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EVERYONE: solve for x:9(x – 2) – 4x = 2(2x + 1) + 1

9x – 18 – 4x = 4x + 2 + 1

5x – 18 = 4x + 3

x – 18 = 3

x = 21

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Example: (1/2)x + 5/4 = 7/4

4[(1/2)x + (5/4)] = 4[7/4]

4[(1/2)x] + 4[5/4] = 4[7/4]

2x + 5 = 7

2x = 2

x = 1

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25

Example: 0.3x + 1.4 = 2.25x – 9.02

100[0.3x] + 100[1.4] = 100[2.25x] – 100[9.02]

30x + 140 = 225x – 902

140 = 195x – 902

1042 = 195x

1042/195 = x

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Special Cases

• Example: 2x + 3 = 3 + 2x2x + 3 – 2x = 3 + 2x – 2x3 = 3

Example: x + 3 = x – 5x + 3 – x = x – 5 – x3 = -5

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Solving for a Variable

• Example: solve a + b = c for a

a + b – b = c – ba = c – b

Example: solve A = (1/2)bh for h2*A = 2*(1/2)bh2A = bh2A/b = bh/b2A/b = h

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Translating to Math

• Ex. three plus a number 3 + x

Ex. ten more than a number N + 10

Ex. 9 minus a number 9 – x

Ex. 20 decreased by an unknown number 20 – n

***Ex. 4 less than a number x – 4

Ex. 4 times a number 4 * x OR 4x

Ex. a number times a different number x * y OR xy.

Ex. 7 divided by a number 7/x

Ex. A number divided by 2 n/2

Ex. A number squared increased by six x2 + 6

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Page 139 #34• PetSmart has a sale offering 10% off of all

pet supplies. If Amanda spent $15.72 on pet supplies before tax, what was the price of the pet supplies before the discount?

• Name the price before discount x.• x - x * 0.10 = 15.72• x - 0.10x = 15.72• 0.9x = 15.72• x is about $17.47

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Page 140 #46• A bookcase with three shelves is built by a student. If the height of the

bookcase is to be 2 ft longer than the length of a shelf and the total amount of wood to be used is 32 ft, find the dimensions of the bookcase.

• Let x = width (length of shelf) and let x + 2 = height

• From picture in book, there are 4 pieces of wood for width and 2 pieces of wood for the height.

• 4x + 2(x + 2) = 32

• 4x + 2x + 4 = 32

• 6x + 4 = 32

• 6x = 28

• x = 28/6

• x = 14/3 = 4 2/3

• So, width of bookcase is 4 2/3 ft and height is 6 2/3 ft.30