Welcome to Survey of Mathematics

Unit 4 – Variation, Inequalities & Graphs

Agenda• Miscellaneous

Administrative “stuff”• Direct and Inverse Variation• Inequalities & Number Line

Graphs• Graphing in the (x, y)

system• Wrap up

Variation• If “y varies directly as x” it means

y = k x• Example: The amount of interest earned on

an investment, I, varies directly as the interest rate, r. If the interest earned is $50 when the interest rate is 5%, find the amount of interest earned when the interest rate is 7%.

Here’s the variation Equation: I = r x Substitute for I and r: $50 = 0.05x Solve for x: 1000 = x

Variation• Here’s the variation Equation: I = r x Substitute for I and r: $50 = 0.05x Solve for x: 1000 = xNow let’s find the Interest we earn using what

we just found:Substituting x = $1000, r = 7%

I = rxI = 0.07(1000)I = $70

The amount of interest earned is $70.

Variation• If “y varies inversely as x” it

means y = k/x• Suppose y varies

inversely as x. If y = 12 when x = 18, find y when x = 21.

y k

x

12 k

18216 k

• Now substitute 216 for k, and find y when x = 21.

216

2110.3

ky

x

y

y

Inequalities

• Less than: a < b• Greater than: a > b• Less than or equal to: a < b • Greater than or equal to: a > b

*Note: for the purposes of typing inequalities on the discussion board, use <= and >= for less than or equal to and greater than or equal to respectively.

Inequalities

• Basic Rules for using inequality symbols:

• If you add or subtract on one side of an inequality, you must add or subtract in the same way on the other:– Example: x + 1 > 5

x + 1 - 1 > 5 – 1 x > 4

Inequalities

• Basic Rules for using inequality symbols:

• If you multiply or divide on both sides by a positive number, you must do the same on both sides:– Example: 2x <= 12

2x/2 <= 12/2 x <= 6

Inequalities

• Basic Rules for using inequality symbols:

• If you multiply or divide by a negative number on both sides, you must also switch the direction of the inequality:– Example: (-1/3)x < 5

(-3)*(-1/3)x > (-3)(5) x > -15

Inequalities

EXAMPLE: Graph the solution to the x > -2

To graph this solution on the number line, use a “filled in dot” on -2 and darken in the number line everywhere to the right of -2

EXAMPLE: Graph the solution to x < 5

To graph this solution on the number line, use a “open dot” on 5 and darken in the number line everywhere to the right of 5

Inequalities

• Solve - 3x - 5 < 16; write the solution set in interval notation.

- 3x - 5 < 16- 3x - 5 + 5 < 16 + 5 Add 5 to both sides

-3x < 21 -3x / -3 > 21 / -3

Remember to “flip” the inequality when dividing by a negative

x > -7Solution on the number line:

The Rectangular Coordinate System

Graphing Linear Equations

Given the equation y = x + 2 and x = 2, find y.Answer:We would substitute the value given for x and solve for y.y = (2) + 2y = 4

What is the x coordinate of y = x + 2 if y = 0? (__, 0)

The ordered pair that satisfies this equation would be (2, 4). Why?

Graphing Linear Equations

• Graph of 3x – y = 3 is shown at right

• The x-intercept:– Let y = 0, find x

• The y-intercept:– Let x = 0, find y

• Linear equations are those that may be written in the form Ax + By = C

The Slope of a Line

• Slope of a line: “rise over run”EXAMPLE: the slope of a line that passes through

the points (1, 2) and (5, 5)

Slope = Difference in y ’s Difference in x ’s

Slope = 5 - 2 5 – 1

Slope = 3 4

(1, 2)

(5, 5)

Wrap Up

• Access notes via Doc Sharing• Variation• Inequalities• Graphs of Lines

• See you on the Discussion Board!