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 = x Now let’s find the Interest we earn using what we just found: Substituting x = $1000, r = 7% I = rx I = 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. Now substitute 216 for k, and find y when x = 21.

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 = 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 > 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 (-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 < Add 5 to both sides -3x < 21 -3x / -3 > 21 / -3 Remember to “flip” the inequality when dividing by a negative x > -7 Solution 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) + 2 y = 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 = – 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!