Summer MA Lesson 14 Section 1.7 (part 2) and Sections 1.1 & 2.8

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1 Summer MA 1500 Lesson 14 Section 1.7 (part ) and Sections 1.1 &.8 I Solving Absolute Value Inequalities Absolute Value Inequalities: u < c or u c, if c 0 The inequalit u < cindicates all values less than c units from the origin. Therefore < c is equivalent to the compound inequalit c< u < c. There is a similar statement u for u c. c c Absolute Value Inequalities: u > c, u c, if c> 0 The inequalit u > cindicates all values more than c units from the origin. Therefore u > c is equivalent to the inequalit statement u < c or u > c. There is a similar statement for u c. c c To help ou keep the two cases straight in our head, I recommend thinking of a number line. If the absolute value is greater than a positive number c, it is greater than that man units awa from zero. u u -c c If the absolute value is less than a positive number c, it is within that man units of zero. u -c c E 1: Solve each. Write solutions using interval notation and graph the solutions on a number line. Hint: Alwas isolate the absolute value before writing an inequalit without the absolute value. a) + 4 < 6 1

2 b) c) < 1 3 E : Solve each inequalit. Write the solutions using interval notation and graph the solutions on a number line. a) > 5 b) c) > 7

3 II Applied Problems E 3: Mar wants to spend less than $600 for a DVD recorder and some DVDs. If the recorder of her choice costs $45 and DVDs cost $7.50 each, how man DVDs could Mar bu? E 4: The percentage, P, of US voters who used punch cards or lever machines in national elections can be modeled b the formula P = where is the number of ears after In which ears will fewer than 35.7% of US voters use punch cards of lever machines? E 5: A college provides its emploees with a choice of two medical plans shown in the following table. Plan 1: $100 deductible pament 30% of the remaining paments Plan : $00 deductible pament 0% of the remaining paments For what size hospital bills is plan better for the emploee than plan 1? (Assume the bill is over $00.) 3

4 E 6: The room temperature in a public courthouse during a ear satisfies the inequalit T 71 < 3 where T is in degrees F. Epress the range of temperatures without the absolute value smbol. III Rectangular Coordinate Sstem Rectangular Coordinate Sstem: II I There are 4 quadrants, represented b Roman Numerals counterclockwise from the upper right. Ever point on a rectangular coordinate sstem is represented b an ordered pair, (, ). and are called the coordinates of the point. III origin IV -ais E 7: -ais a) In which quadrant(s) do the coordinates of a point have the same sign? b) In which quadrant would the point (-, 3) be found? c) The point (1, 0) is found on which ais? d) What point would be 5 right and 6 down from the origin? IV Graphs of Equations An equation in variables can be represented on a rectangular coordinate sstem b plotting points (ordered pairs) that satisfies the equation. The complete graph contains all ordered pairs whose coordinates satisf the equation. 1. Make a table b selecting a value for and solving for (or vice-versa).. Plot enough points to be able to sketch a smooth curve or line to represent the graph. 4

5 E 8: Graph the following equations. Use the values of 3,, 1,0,1, and 3 for. a) = 3 1 b) = 5

6 1 c 3 ) = + 1 V Intercepts An -intercept of a graph is an -coordinate of a point where the graph intersects the - ais. The -intercept of a graph is the -coordinate of a point where the graph intersects the -ais. Yes, our tetbook uses single numbers to describe intersepts. Some tetbooks use the complete ordered pair to define the intercepts. (See the stud tip at the bottom of page 94 of the tetbook.) Since an intercept alwas lies on an ais, the coordinate other than the one given as the -intercept or -intercept is zero. Zero is our friend! E 9: Use the following graphs to identif the intercepts. a) 6

7 b) ( 0,4) (0,5) ( 4,0) ( 1,0 ) (,0) (, 3) c) d) 7

8 VI Applied Problems On page 96 of our tetbook is a graph showing the probabilit of divorce b the wife s age at the time of marriage. The two mathematical models that approimate the data displaed in the graphs are d = 4n+ 5 (where d = the percentage of marriages ending in divorce for n ears after the marriage for a wife under 18 at time of marriage) and d =.3n+ 1.5 (for a wife over 5 at time of marriage). E 10: a) Use the graph to approimate the percentage of marriages ending in divorce after 5 ears of marriage if the wife was under 18. b) Use the correct model (formula) to approimate the percentage of marriages ending in divorce after 5 ears of marriage if the wife was under 18. c) Does the value given b the mathematical model underestimate or overestimate the actual percentage of marriages ending in divorce after 5 ears for a bride under 18 shown in the graph? B how much? VII Distance Between Points on a Coordinate Sstem Notation for points: Points are labeled using capital letters. Sometimes a point ma be written as follows: P (, 1 1). This is read 'point P with coordinates of sub 1 and sub 1. To find the distance between two points, the Pthagorean Theorem could be used. P( ), d 1 - Q, ) ( 8

9 Distance Formula: d = ( ) + ( ) 1 1 d = ( ) + ( ) 1 1 The second line is known as the distance formula between two points. E 11: Find the eact distance (in simplified form) between the given points. a) P(0,5), Q(6,-3) b) P(3, 3), Q( 5,5) E 1: Approimate the distance between the given points to the nearest hundredth..6, 3.1, ( 8.5,.1) ( ) Midpoint Formula: The midpoint of two point P and Q is the point midwa between P and Q. Its coordinates are the average of the coordinates of P and Q and the average of the coordinates of P and Q The midpoint of P ( 1, 1) and Q(, ) is M,. Stud Tip page 95: The midpoint requires sum of the coordinates. The distance formula requires difference of the coordinates. E 13: Find the midpoint of each pair of points. a) P(0,5), Q(6, 3) 3 b) P(4, 3), Q 5, 9

10 (optional) E 14: Determine is a triangle with the following vertices is an equilateral triangle. A ( 13, ), B(9, 8), C(5, ) 10