2.4 Exercises. Interval Notation Exercises 1 8: Express the following in interval notation. Solving Linear Inequalities Graphically

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1 _chpp7-8.qd //8 4: PM Page 4 4 CHAPTER Linear Functions and Equations.4 Eercises Interval Notation Eercises 8: Epress the following in interval notation Ú { 8}. { - 4} 7. { } 8. { 7 } Solving Linear Inequalities Smbolicall Eercises 9 8: Solve the inequalit smbolicall. Epress the solution set in set-builder or interval notation Ú ( - ) + 7. ( + ) t +. Ú (z - 4) Ú ( - z) (z - ) + z Ú ( + ). - ( - ) -. 4t t t t ( - z) - z + z Ú z - Solving Linear Inequalities Graphicall Eercises 9 4: (Refer to Eample.) Solve the inequalit graphicall. Use set-builder notation Ú Eercises 47 : Use the given graph of = a + b to solve each equation and inequalit. Write the solution set to each inequalit in set-builder or interval notation. (a) a + b = (b) a + b (c) a + b Ú = t z + ( - z) - 4 z Ú 4 (z - ) + z = Ú Ú 4 - ( + ) Ú ( + 4) ( + ) Ú = = ( - ) +

2 _chpp7-8.qd //8 4: PM Page 4.4 Linear Inequalities 4 Eercises 4: -Intercept Method (Refer to Eample 4.) Use the -intercept method to solve the inequalit. Write the solution set in set-builder or interval notation. Then solve the inequalit smbolicall Eercises : Solve the linear inequalit graphicall. Write the solution set in set-builder notation. Approimate endpoints to the nearest hundredth whenever appropriate Use the figure to solve each equation or inequalit. (a) ƒ() = g() (b) g() = h() (c) ƒ() g() h() (d) g() 7 h() 7 = g() = h() 4 = f () ( - 99) + Ú ( -.) - ( +.) ( ) Solving Linear Inequalities Numericall Eercises 9 and 7: Assume represents a linear function with the set of real numbers for its domain. Use the table to solve the inequalities. Use set-builder notation. 9. 7, 7., Ú Eercises ; Solve the compound linear inequalit graphicall. Write the solution set in set-builder or interval notation, and approimate endpoints to the nearest tenth whenever appropriate The graphs of two linear functions ƒ and g are shown. (a) Solve the equation g() = ƒ(). (b) Solve the inequalit g() 7 ƒ() X 4 X Y Eercises 7 78: Solve each inequalit numericall. Write the solution set in set-builder or interval notation, and approimate endpoints to the nearest tenth when appropriate Ú X 4 X Y ( - p) -..( -.7) +. = g() = f() (8, 7) You Decide the Method Eercises 79 8: Solve the inequalit. Approimate the endpoints to the nearest thousandth when appropriate p ( -.) p Ú -.7

3 _chpp7-8.qd //8 4: PM Page CHAPTER Linear Functions and Equations Applications 8. Distance between Cars Cars A and B are both traveling in the same direction. Their distances in miles north of St. Louis after hours are computed b the functions ƒ A and ƒ B, respectivel. The graphs of ƒ A and ƒ B are shown in the figure for. (a) Which car is traveling faster? Eplain. (b) How man hours elapse before the two cars are the same distance from St. Louis? How far are the from St. Louis when this occurs? (c) During what time interval is car B farther from St. Louis than car A? Distance (miles) Distance Function ƒ computes the distance in miles between a car and the cit of Omaha after hours, where. The graphs of ƒ and the horizontal lines = and = are shown in the figure. (a) Is the car moving toward or awa from Omaha? Eplain. (b) Determine the times when the car is miles and miles from Omaha. (c) Determine when the car is from to miles from Omaha. (d) When is the car s distance from Omaha greater than miles? = f A () (., ) Time (hours) = f B () 8. Clouds and Temperature (Refer to Eample.) Suppose the ground-level temperature is F and the dew point is F. (a) Use the intersection-of-graphs method to estimate the altitudes where clouds will not form. (b) Solve part (a) smbolicall. 8. Temperature and Altitude Suppose the Fahrenheit temperature miles above ground level is given b the formula T() = 8-9. (a) Use the intersection-of-graphs method to estimate the altitudes where the temperature is below freezing. Assume that the domain of T is. (b) What does the -intercept on the graph of represent? (c) Solve part (a) smbolicall. = T() 87. Prices of Homes The median prices of a single-famil home in the United States from 99 to can be approimated b the formula P() = ,, where = corresponds to 99 and = to. (Source: National Association of Realtors.) (a) Interpret the slope of the graph of P. (b) Estimate the ears when the median price range was from $4, to $94,. 88. Population Densit The population densit D of the United States in people per square mile during ear from 9 to can be approimated b the formula D() =.8-8. (Source: Bureau of the Census.) (a) Interpret the slope of the graph of D. (b) Estimate when the densit varied between and 7 people per square mile. 89. Broadband Internet Connections The number of households using broadband Internet connections, such as cable and DSL, increased from million in to million in 4. (Source: emarketer.) (a) Find a linear function given b Distance (miles) = f () = = 4 B() = m( - ) + that models these data, where is the ear. (b) Use B() to estimate the ears when the number of households using broadband Internet connections was 4 million or more. Assume that the domain of B is to. Time (hours)

4 _chpp7-8.qd //8 4: PM Page 4.4 Linear Inequalities 4 9. Online Betting Consumer gambling losses from online betting were $4 billion in and $ billion in. (Source: Christiansen Capital Advisors.) (a) Find a linear function given b B() = m( - ) + that models these data, where is the ear. (b) Use B() to estimate the ears when consumer losses from online betting were more than $ billion. Assume that the domain of B is to Consumer Spending In consumers used credit and debit cards to pa for 4% of all purchases. This percentage is projected to be % in. (Source: Bloomburg.) (a) Find a linear function P that models the data. (b) Estimate when this percentage was between 4% and %. 9. VISA Cards Annual transactions on VISA cards increased from $4 billion in to $ billion in 7. (Source: CardWeb.) (a) Find a linear function V that models the data. (b) Estimate when this number was between $4 billion and $4 billion. 9. Modeling Sunrise Times In Boston, on the 9th da (March ) of 8 the sun rose at : A.M., and on the 9th da (Ma 8) the sun rose at : A.M. Use a linear function to estimate the das when the sun rose between :4 A.M. and : A.M. Do not consider das after Ma 8. (Source: R. Thomas.) 94. Modeling Sunrise Times In Denver, on the 77th da (March 7) of 8 the sun rose at 7: A.M., and on the th da (April ) the sun rose at : A.M. Use a linear function to estimate the das when the sun rose between : A.M. and :4 A.M. Do not consider das after April. (Source: R. Thomas.) r 9. Error Tolerances Suppose that an aluminum can is manufactured so that its radius r can var from.99 inches to. inches. What range of values is possible for the circumference C of the can? Epress our answer b using a three-part inequalit. 9. Error Tolerances Suppose that a square picture frame has sides that var between 9.9 inches and. inches. What range of values is possible for the perimeter P of the picture frame? Epress our answer b using a threepart inequalit. 97. Modeling Data The following data are eactl linear (a) Find a linear function ƒ that models the data. (b) Solve the inequalit ƒ() Modeling Data The following data are eactl linear (a) Find a linear function ƒ that models the data. (b) Solve the inequalit ƒ() 8. Linear Regression 99. Cell Phone Subscribers The table lists the number N of cell phone subscribers worldwide in millions for selected ears. 4 N Source: CTIA The Wireless Association. (a) Find a linear function N that models the data. (b) Estimate the ears when this number was from 4 million to 8 million. (c) Did our estimate involve interpolation or etrapolation?. Home Ownership Rates The table lists the percentage P of U.S. homes that are owned b their occupant rather than rented for selected ears P 47% % 4% 9% Source: Bureau of the Census. (a) Find a linear function P that models the data. (b) Estimate the ears when this percentage was from 8% to %. (c) Did our estimate involve interpolation or etrapolation?

5 _chpp7-8.qd //8 4: PM Page 4 4 CHAPTER Linear Functions and Equations Writing about Mathematics. Suppose the solution to the equation a + b = with a 7 is = k. Discuss how the value of k can be used to help solve the linear inequalities a + b 7 and a + b. Illustrate this process graphicall. How would the solution sets change if a?. Describe how to numericall solve the linear inequalit a + b. Give an eample.. If ou multipl each part of a three-part inequalit b the same negative number, what must ou make sure to do? Eplain b using an eample. 4. Eplain how a linear function, a linear equation, and a linear inequalit are related. Give an eample. EXTENDED AND DISCOVERY EXERCISES. Arithmetic Mean The arithmetic mean of two numbers a + b a and b is given b. Use properties of inequalities to show that if a b, then a a + b b.. Geometric Mean The geometric mean of two numbers a and b is given b ab. Use properties of inequalities to show that if a b, then a ab b. CHECKING BASIC CONCEPTS FOR SECTIONS. AND.4. Solve the linear equation 4( - ) = ( - ) - b (b) using each method. Compare our results. (a) Graphical (b) Numerical (c) Smbolic (c). Solve the inequalit ( - 4) 7 -. Epress the solution set in set-builder notation.. Solve the compound inequalit - -. Use set-builder or interval notation. 4. Use the graph to the right to solve each equation and inequalit. Then solve each part smbolicall. Use setbuilder or interval notation when possible. (a) -( - ) - - = -( - ) ( - ) - - = ( ) Absolute Value Equations and Inequalities Evaluate and graph the absolute value function Solve absolute value equations Solve absolute value inequalities Introduction A margin of error can be ver important in man aspects of life, including being fired out of a cannon. The most dangerous part of the feat, first done b a human in 87, is to land squarel on a net. For a human cannonball who wants to fl 8 feet in the air and then land in the center of a net with a -foot-long safe zone, there is a margin of error of feet. That is, the horizontal distance D traveled b the human cannonball can var between 8 - = feet and 8 + = feet. (Source: Ontario Science Center.) This margin of error can be epressed mathematicall b using the absolute value inequalit ƒ D - 8 ƒ. The absolute value is necessar because D can be either less than or greater than 8, but b not more than feet.