Welcome to the Unit 4 Seminarfor Survey of Mathematics!

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

Unit 4 Seminar Agenda

• 4.1 Variation

• 4.2 Linear Inequalities

• 4.3 Graphing Linear Equations

3

Direct Variation

• If a variable y varies directly with a variable x, then y = kx, where k is the constant of proportionality.

3

4

The resistance, R, of a wire varies directly as its length, L.If the resistance of a 30 ft length of wire is 0.24 ohm,

determine the resistance of a 40 ft length of wire?

• R = kL

• 0.24 = k(30)

• 0.24/30 = k

• 0.008 = k

Now that k is determined, we can find R if L=40:

• R = 0.008L

• R = 0.008(40)

• R = 0.32 ohm4

5

Inverse Variation

• If a variable y varies inversely with a variable x, then y = k/x, where k is the constant of proportionality.

5

6

The time, t, for an ice cube to melt is inversely proportional to the temperature, T, of the water in which the ice cube is

placed. If it takes an ice cube 2 minutes to melt in 75 degree F water, how long will it take an ice cube of the

same size to melt in 80 degree F water?

• t = k/T

• 2 = k/75

• 150 = k

Now that k is determined, in the second part of the problem, we are now looking for t when T=80.

• t = 150/T

• t = 150/80

• t = 1.875 minutes

Always check if the answer is “reasonable”. 6

7

Joint Variation

• The general form of a joint variation, where y varies directly as x and z, is y = kxz, where k is the constant of proportionality.

7

8

The volume, V, of a pyramid varies jointly as the area of its base, B, and height, h. If the volume of a pyramid is 12

cubic feet when the area of the base is 4 square feet and the height is 9 feet, find the volume of a pyramid when the area of the base is 16 square feet and the height is 9 feet.

• V = kBh

• 12 = k(4)(9)

• 12 = 36k

• 12/36 = k

• 1/3 = k

• V = (1/3)Bh

• V = (1/3)(16)(9)

• V = 48 cubic feet 8

9

EVERYONE: C varies inversely as J. If C = 7 when J = 0.7, determine C when J = 12.

10

• C = k/J

10

EVERYONE: C varies inversely as J. If C = 7 when J = 0.7, determine C when J = 12.

11

• C = k/J• 7 = k/0.7• 4.9 = k

11

EVERYONE: C varies inversely as J. If C = 7 when J = 0.7, determine C when J = 12.

12

• C = k/J• 7 = k/0.7• 4.9 = k• Now that we know k• C = k/J• C = 4.9/12• C = 0.4083

12

EVERYONE: C varies inversely as J. If C = 7 when J = 0.7, determine C when J = 12.

13

Inequality signs< is less than

ex: -3 < 5

≤ is less than or equal to

ex: x ≤ 0

> is greater than

ex: 5 > 1

≥ is greater than or equal to

ex: x ≥ 9

13

14

4x + 9 > 25

• Use the same principles you learned with equations (except for one caveat).

• 4x + 9 - 9 > 25 - 9

• 4x > 16

• x > 4

14

15

10 - 3x ≤ 21

• 10 - 3x - 10 ≤ 21 - 10

• -3x ≤ 11

• -3x/-3 ≥ 11/-3 (this is the exception: notice the change in direction of the inequality)

• x ≥ -11/315

16

-(1/4)x - 5 > 9

• -(1/4)x - 5 + 5 > 9 + 5

• -(1/4)x > 14

• (-4)(-1/4)x < (-4)(14) (note change in inequality sign)

• x < -5616

17

56 > -9x + 2 > 29

• 56 - 2 > -9x + 2 - 2 > 29 - 2

• 54 > -9x > 27

• 54/-9 < -9x/-9 < 27/-9 (sign change again!)

• -6 < x < -3 (note that numbers would not make sense if you “forget” to change direction) -6 > x > -3 is not true 17

18

Coordinate Grid

x axis

y axis

origin

Quadrant IQuadrant II

Quadrant III Quadrant IV

A

B

C

D

19

Example: x + y = 5. x = 0 and y = 5 gives ordered pair (0,5)

x = 1 and y = 4 gives ordered pair (1, 4)

x = 2 and y = 3 gives ordered pair (2, 3)

x = 3 and y = 2 gives ordered pair (3,2)

x = 4 and y = 1 gives ordered pair (4, 1)

x = 5 and y = 0 gives ordered pair (5, 0)

20

Graphing Linear Equationsx + y = 5

21

Example: find three points on the line 2x – 3y = 12

First point: I will replace x with 0.2(0) – 3y = 120 – 3y = 12-3y = 12-3y/(-3) = 12/(-3)y = -4

Ordered pair is (0, -4)

21

22

Example: find three points on the line 2x – 3y = 12

Second point: I will replace x with 2.2(2) – 3y = 124 – 3y = 124 – 3y – 4 = 12 – 4-3y = 8-3y/(-3) = 8/(-3)y = -8/3Ordered pair is (2, -8/3)

22

23

Example: find three points on the line 2x – 3y = 12

Third point: I will replace x with -3.2(-3) – 3y = 12-6 – 3y = 12-6 – 3y + 6 = 12 + 6-3y = 18-3y/(-3) = 18/(-3)y = -6Ordered pair is (-3, -6)

23

24

(0, -4)(2, -8/3)(-3, -6)

24

Graph of 2x -3y = 12

25

4x + y = -1

− − − − − − − − −

−

−

−

−

−

−

−

−

−

−

26

Example: What are the x- and y-intercepts of 3x – 7y = 21?

To find the x-intercept,

let y = 0.

3x – 7(0) = 21

3x – 0 = 21

3x = 21

x = 7

(7,0)

To find the y-intercept, let x = 0.

3(0) – 7y = 21

0 – 7y = 21

-7y = 21

y = -3

(0, -3)

Positive Slope

As x values increase, y values also increase

Negative Slope

As x values increase, y values decrease

Horizontal Line

Form: y = constant number

Vertical Line

Form: x = constant number

Finding Slope Graphically

(-6, -1)

(2, 4)Up 5

Right 8

Slope is 5/8

Finding Slope Graphically

(-4, 1)

(3, -3)Right 7

Down 4

Slope is -4/7

Finding slope given 2 pointsFind the slope between (1,3) and (4,5)

m = 2

3

Example:Find the slope of the line that passes through the points (-2, 6) and (-1, -2).

EVERYONE: Find the slope of the line that passes through the points

(0, 3) and (7, -2)

EVERYONE: Find the slope of the line that passes through the points

(0, 3) and (7, -2)

m = -2 – 3

7 – 0

m = -5

7

ExampleFind the slope of the line that passes

through the points (1, 7) and (-3, 7).

m = 0

Examplefind the slope of the line that passes

through the points (6, 4) and (6, 2).

m = UNDEFINED

(Division by zero is undefined)

Finding slope and y-intercept given an equation

• First solve for y• When you have it in the form y = mx + b

then m is your slope and b is your y-intercept

3x + y = 7

y = -3x + 7

m = -3 is your slope

b = (0, 7) is your y-intercept

y = -3x + 7

ExampleFirst move your 2x to the other side

simplify

We need to isolate the y so divide both sides by 4

simplify