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MM150 Unit 3 Seminar Agenda Sections

Examples Variables: x, y, z, a Algebraic Expression: a + b 4x – 7 6y x/4 They can be longer, like these: 3x 2 – 7y z – 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 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 This statement is false. 5

6 Evaluating Expressions Exponents: x 2 AND 3 4 AND -7y 3 AND 5 9 2*2*2*2*2*2*2, you can rewrite this as 2 7 x*x*x*x is x 4 (2a)(2a)(2a) is (2a) 3 (x + 6)(x + 6) is (x + 6) 2 x^2 is the same as x 2 2^3 = 2 3 = 2*2*2 = 8 Be careful! (-2) 4 = (-2)(-2)(-2)(-2) = = -(2*2*2*2) = -16 6

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 = = 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.

8 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 8

9 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. 9

10 Terms Examples of terms: Constants: 3, -5, 0, 1/7, Pi Variables: a, b, c, x, y, z Products: 3x, ab 2, -99ay 5 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): x 2 + 7x - 10 Expressions can have four terms or more (polynomial): x 2 y + xy - 11y + 23 NOTE: Decreasing power of the variable. 10

11 Like and Unlike Terms 5x and 3x are like terms 6ab and -9ab are like terms 16x 2 and x 2 are like terms -0.35ac 5 and -400ac 5 are like terms You can simplify like terms! For example, 12a + 4a = 16a 57x – 33x = 24x 9x 2 + 3x 2 + x 2 = 13x 2 -ab + (-4ab) = -5ab You cannot simplify unlike terms!! 2x + 2y + 3x = 5x + 2y 8x 2 – 4x + x 2 = 9x 2 – 4x 11

12 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 = Here we only added 7 to one side Here we added 2 to both sides 12

13 Solving Equations x - 7 = 18 x = x = = -4 + x = -4 + x = x EVERYONE: 6 = x What is x? 13

14 Example: x = 11 – x = x – 11 = 9 – 11 x = -2 14

15 EVERYONE: solve for x: 2 – 8 = x – 5 – 1 -6 = x – 6 0 = x 15

16 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 16

17 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 = -1/2 6(x/6) = 6(-1/2) x = -6/2 x = -3 17

18 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 18

19 Solving Equations Example: -3x = 18 -3x/(-3) = 18/(-3) x = -6 Example: 9x = -8 9x/9 = -8/9 x = -8/9 Example: -x = -3 -1(-x) = -1(-3) -OR- -x/(-1) = -3/(-1) x = 3 19

20 3 = 15x = 15x -17/15 = x 20 Example: 3 – 12x = 3x + 20

21 EVERYONE: solve for x: – 6x = 2x + x – 6x = 3x = 9x = 9x 14/9 = x 21

22 Example: 3(2x – 5) – 7 = x(x + 4) – x 2 6x – 15 – 7 = x 2 + 4x – x = -2x 11 = x 22

23 EVERYONE: solve for x: 9(x – 2) – 4x = 2(2x + 1) + 1 9x – 18 – 4x = 4x x – 18 = 4x + 3 x – 18 = 3 x = 21 23

24 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 24

25 Example: 0.3x = 2.25x – [0.3x] + 100[1.4] = 100[2.25x] – 100[9.02] 30x = 225x – = 195x – = 195x 1042/195 = x 25

26 Special Cases Example: 2x + 3 = 3 + 2x 2x + 3 – 2x = 3 + 2x – 2x 3 = 3 Example: x + 3 = x – 5 x + 3 – x = x – 5 – x 3 = -5 26

27 Solving for a Variable Example: solve a + b = c for a a + b – b = c – b a = c – b Example: solve A = (1/2)bh for h 2*A = 2*(1/2)bh 2A = bh 2A/b = bh/b 2A/b = h 27

28 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 x

29 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 = x x = x = x is about $

30 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