Welcome to MM150! Unit 3 Seminar To resize your pods: Place your mouse here. Left mouse click and hold. Drag to the right to enlarge the pod. To maximize chat, minimize roster by clicking here
Jan 02, 2016
Welcome to MM150!
Unit 3 Seminar
To resize your pods: Place your mouse here.
Left mouse click and hold.Drag to the right to enlarge the pod.
To maximize chat, minimize roster by clicking here
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
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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
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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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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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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