4. Linear Inequality Word Problems[1]

Linear inequality word problems 1. The followi following ng function function gives gives the amount amount of of money owed owed on a short short term term loan after after t weeks. weeks. 1

 A ( t )= 100 1.25 2 ⋅

Which numerical expression best approximates the annual interest rate, excluding any late fees !1 year"#$ weeks.% 26 &. 1−0.25 '.   0.2526−1 (. $)*+.$# . $) * 1.$# (orrect answeranswer- ' ifficulty level- $ $. The rabbi rabbitt pop popula ulatio tion n in an isolated isolated forest forest rises rises and falls falls dependin depending g on the populat population ion of   predators. Within Within the year $+1, the population, p, in thousands of rabbits mmm months after  /anuary 1, $+1 is p"+.+#!m01.#%!m0.#%21+  p"+.+#!m01.#%!m0.#%21 + The population reached ten thousand some time in 3ebruary. &t what other time, given as months after /anuary 1, $+1, did the rabbit population reach ten thousand (orrect answeranswer- .# ifficulty level- $ 4. The 5hang 5hang time5 time5 of a footba football ll is the amount amount of time the footba football ll stays stays in the air air after after being being kicked. The height, in meters, of the football above the ground at ti me t can be modeled by the quadratic function-

( )

2

h t  =−4.9 t  + 19.6 t 

Which of the following equivalent expressions displays the hang time of the football as a constant or coefficient coefficient &. −4.9 (t −2 )2+ 19.6 '. (. .

−4.9 t (t − 4 ) −4.9 ( t −3 )2−9.8 t + 44.1 −4.9 ( t −1 )2+ 9.8 t +4.9

(orrect answeranswer- ' ifficulty level- $ .

6aia throws throws a stone stone vertically vertically upward upward from a bridge. bridge. The The height, height, in meters, meters, of the stone above above the water at time t can be modeled by the quadratic function2

)=−4.9 t  +9.8 t + 39.2 h ( t )=− &fter how many seconds does the stone hit the water &. 1 second '. $ se seconds (.  se seconds .  seconds (orrect answeranswer- ( ifficulty level- $ #.

7ore than than #+ students students trave traveled led to a state state park park for a field field trip. The The school school allowed allowed ) students students to travel by car, and the rest traveled on 11 buses, each of which held the same number of  students. 8f there were s students in each bus, which inequality best represents this situation &. 11 s + 6 > 450 '. 11 s + 6 < 450 6 s + 11> 450 (. . 6 s + 11< 450 (orrect answer- & ifficulty ifficult y level- $

).

& barber barber charg charges es 91$ for a haircut. haircut. :is operati operating ng expenses expenses are, are, on average, average, 94; per day. day. :e calculates his profit by subtracting his operating costs from the money he earns from the haircuts he gives. 8n a given day, the barber expects to make a profit of at least 9). 8f the  barber gives h haircuts haircuts in a day, which inequality best models models this situation 12 h− 37 ≥ 86 &. '. 12 (h −37 ) ≥ 86 12 h + 37 ≥ 86 (. . 12 (h + 37 ) ≥ 86 (orrect answer- & ifficulty ifficult y level- $

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& distrib distributor utor ships ships  miles. &fter he gets off the bus, he can choose one of several walking routes to school. The total amount of miles he travels, via walking and bus, does not exceed #.. What is the greatest possible number of miles he walks after getting off the b us &. +.+ miles '. +.4 mi miles (. +.> mi miles . 1.# miles (orrect answeranswer- ( ifficulty level- 4 1. 6arim 6arim has a 94# gift card card to his favorite favorite restaura restaurant. nt. 7eal tax in his area is 1.4C. 6arim 6arim would also like to leave a 9# tip. 8f c represents 6arim@s pre?tax bill, and he wants to pay the entir entiree meal meal amoun amount, t, includ including ing tip and tax, tax, using using the gift card, card, which which of the followin following g inequalities best models the situation described above &.   1.143 c −5 ≥ 35 '.   1.143 c + 5 ≤ 35 (.   0.143 c −5 ≤ 35 .   0.143 c + 5 ≥ 35 (orrect answeranswer- ' ifficulty level- 4 1#. Ane of the rules in a public public speaking speaking contest requires requires contesta contestants nts to speak for as close to 4 minutes !1+ seconds% as possible. (ontestants lose $ points for each second they speak either  over or under the 4 minutes. 8f a contestant lost 1 points, what are the possible lengths of  time they spoke &. 1;1 1;1 and and 1) 1) sec secon onds ds '. 1; 1; and and 1) 1) sec secon onds ds (. 1;1 1;1 and and 1> 1> sec secon onds ds . 1; 1; and and 1> 1> sec secon onds ds

(orrect answeranswer- ( ifficulty level- 1 1). & gemologis gemologistt determines determines that a gemstone gemstone weighs 1.$4# 1.$4# carats. Duidelines Duidelines allow a maximum weighi weighing ng error error of +.++1 +.++1 carats carats in repor reportin ting g gemst gemstone one weight weight.. Which Which of the follow following ing equations can be used to determine the smallest and largest possible true weights, c, of the gemstone ¿ c −0.001 ∨¿ 1.235 &. ¿ c −1.235 ∨¿ 0.1235 '. ¿ c −0.001 ∨¿ 0.01 (. ¿ c −1.235 ∨¿ 0.001 . (orrect answeranswer-  ifficulty levellevel- 1 1;. & company company@s @s unit cost for producing producing x units is a minimum when x"$,+++ x"$,+++ units are produced. produced. The unit cost increases 9+.$# for every unit more or less than $,+++ produced. Which of the following following could be the number of units for which the unit cost increases increases by 9;.#+ from the minimum &. 1,;+ 1,;++ + or or $,4 $,4++ ++ '. 1,>; 1,>;# # or $,+$ $,+$# # (. 1,>> 1,>>; ; or $,++ $,++4 4 . 1,>; 1,>;+ + or or $,+ $,+4+ 4+ (orrect answeranswer-  ifficulty levellevel- 1 1. 3elipe has a 1# gallon gallon gas tank in his car. car. :e fills his tank, tank, and then drives the x miles back to college. Epon his arrival, he fills his tank for 9$+.>1. 8f his car averages $ miles per gallon !mpg%, and at the time, gas costs 9$.) per gallon, which of the following best approximates x &. 1$ '. $+ (. 4)+ . #+$ (orrect answeranswer- ' ifficulty level- 1 1>. 8n $++#, approximately approximately 1).#C of seats in in national parliaments were held by women. women. 'etween $++# and $+1, this percentage increased increased at a constant rate of about 4 percentage points every # years. 8f t is the year when approximately 1>C of the seats in national parliaments were held  by women, which of the the following equations best best models the situation &.   16.5 + 3 t =19 3 '.   16.5 + t =19 5 (.   16.5 + 3 ( t −2005 )=19 3 .   16.5 + ( t − 2005 )= 19 5 (orrect answeranswer-  ifficulty levellevel- 1 $+. =lla needs needs a ;+C isopropyl isopropyl alcohol alcohol solution solution as an antiseptic. antiseptic. Fhe has a 1++ milliliter !ml% !ml% container of >+C isopropyl alcohol solution. &pproximately how much water should =lla add to dilute it to a ;+C isopropyl alcohol solution Gound to the nearest tenth. ml of water  (orrect answeranswer- $.) ifficulty level- 1

$1. 3or many trees, trees, the approximat approximatee age of the tree !in years% years% is equal to the diameter diameter of the tree !in inches% times the tree@s growth factor. eniH estimates that a certain red maple tree in her  yard is about + years old. 8f red maple trees have a growth factor of .#, which of the following equations could be used to find x, the diameter of the tree in feet  A. 80= 12( 4.5 x ) 4.5 x 80=  B. 12 12 x 80= C. 4.5  D.

80=

x

( 12)( 4.5 )

(orrect answer- & ifficulty ifficult y level- 1 $$. The property taxes taxes in a town decrease decrease as the distance from from the town hall increases. increases. 3or every every # miles from the town hall, property taxes decrease by +.$ percentage points. Which of the following equations can be used to determine how far north or south of the town hall a house is whose property taxes are +.;) percentage points less than the taxes at the town hall  Iote- y is the position of a house that is directly north or south of the town hall, with yJ+ representing north and yK+ representing south. A.   0.04 ∨ y −5 ∨¿ 0.76 B.   0.04 ∨ y ∨¿ 0.76 C. 4 ∨ y − 5∨¿ 0.76 D. 4 ∨ y ∨¿ 0.76 (orrect answeranswer- ' ifficulty level- 1 $4. /osie@s /osie@s hair currently currently falls about about 1) centimete centimeters rs !cm% below her shoulder shoulder and grows about about # cm every  months. /osie plans to grow her hair and donate about $1 cm of her hair to charity after her next haircut. 8f /osie would like to have her hair fall about # cm below her shoulders after her next haircut, which of the following equations could be used to find the number of  months, mmm, for which /osie must grow her hair A.   1.25 m =5 B.   1.25 m =21 C. 16 + 1.25 m− 21=5 D. 5 + 1.25 m −16 =21 (orrect answeranswer- ( ifficulty level- 1 $. 7elissa has an + inch !in% piece piece of wrought wrought iron that she wishes wishes to use to make make a rectangular  rectangular   piece of art. Fhe wants the length of the rectangular piece of art to be 1$ inches longer than the width. What is the largest length possible &. 1 in '. $) in (. 4 in . ) in (orrect answeranswer- ' ifficulty level- 1 $#. ong ong needs to cut a 4 foot long board into several $ inch lengths. lengths. With each each cut, his saw blade 1 shaves off an additional  inch of the board. What is the maximum number of full $ inch 8 lengths that ong can cut from his board   lengths

(orrect answeranswer- 1; ifficulty levellevel- 1 $). &ria is buying refreshments refreshments for a birthday party. party. Fhe intends to get an ice cream cake for 9$4 and some cupcakes for 9$ each. &ria has a 9# budget but does not have to spend the entire amount. Which inequality best represents the number of cupcakes, c, that &ria can purchase A. c > 11 B. c < 11 (. c ≥ 11 . c ≤ 11 (orrect answeranswer-  ifficulty levellevel- $ $;. Ftan is crochetin crocheting g scarves scarves to sell at a craft craft fair. :e :e will give away the first first $ scarves that he crochets crochets and will sell the rest at the craft fair for 9 each. each. :is scarves scarves are very popular and  people always buy all that he is willing to sell. Ftan@s goal is to earn at least 9;# at the craft fair. 8f n represents the number of scarves which Ftan needs to crochet to meet his goal, which of the following inequalities best models the situation described above 8 n− 2 ≥ 75  A.  B. 8 n ≥ 75 −2 C. 8 ( n−2 ) ≥ 75  D. ( 8 −2) n ≥ 75 (orrect answeranswer- ( ifficulty level- $ $. 8n golf, someone someone who completes completes a course in fewer strokes strokes beats someone someone who completes completes it with more strokes. Mar means the standard number of strokes it takes to complete a specific course. &mateurs &mateurs generally require more strokes. &yumi &yumi averages 1.# times par strokes per game, but her brother aiki averages 1.;# times par  strokes per game. To make the game more even, &yumi adds 1 strokes to her total when she  plays against aiki. &ssuming both &yumi &yumi and aiki play average games, which inequality best represents the par,  p, in a golf game where where &yum &yumii beats her brother aiki aiki A. p > 41 B. p > 54 C. p > 72 D. p > 90 (orrect answeranswer- ( ifficulty level- $ $>. &rianna and :annah :annah are conducting conducting an experiment experiment to test the difference difference threshold threshold for sense of  of  touch on the forehead. 3or trial 1, :annah closes her eyes while &rianna gently touches $  paperclip points 4 millimeters !mm% apart on :annah@s forehead. Then :annah reports whether she could feel 1 or $ points. 3or each of the next trials, &rianna and :annah repeat this process, each time with the points $ more mm apart. :annah reports feeling 1 point until the points are 1#mm apart, when she begins reporting that she feels $ points. Which inequality  best represents the trial numbers in which :annah :annah feels $ points points A. t > 6 B. t < 6 C. t≥7 D. t≤7 (orrect answeranswer- ( ifficulty level- $ 4+. Turkey@s Turkey@s governm government ent requires requires at least least a

2 3

 maNority of Marliament members to approve any

constitutional changes. There are ##+ seats in the Marliament. Af those, 414 of the seats are occupied by members who have pledged to support a particular constitutional change. Which inequality inequality represents represents the number number of additional additional Marliame Marliament nt members, members, m, who would need to support the constitutional change to achieve the required maNority ´ &. m ≥ 53. 6 '. m ≥ 158 (. m ≥ 303 . m ≥ 366. 6´ (orrect answer- & ifficulty ifficult y level- $ 41. 8n one state, state, a minimum of $),++ $),++ valid signatures signatures on a petition petition is required required to get a voter  initiative added to a state?wide ballot. The signature collectors can fit $# signatures per page. They estimate that )1,)++ of the signatures that they collect will not be valid !because of   being duplicates, incomplete, or unregistered voters%. 8f p represents the number of pages of  signatures which the signature collectors should collect in order to get the voter initiative added to the ballot, which of the following inequalities best models the situation described above &. 25 p + 61,600 ≥ 246,400 '. 25 p + 61,600 ≤ 246,400 (. 25 p −61,600 ≥ 246,400 . 25 p −61,600 ≤ 246,400 (orrect answeranswer- ( ifficulty level- $ 4$. When When police police analyH analyHee the broken broken glass glass at a scene, scene, it helps helps them them recon reconstr struct uct the situat situation ion.. :eadlight glass is much less dense than the window glass. The headlight glass will be 1.# grams per milliliter or less. & police officer determines that a fragment of glass displaces 1+.#mLof water. Ese the formula

density =

  mass . Which inequality best shows the volume

 possible mass, m, in grams grams of the fragment if it is headlight glass m≥ 7 &. '. m≤ 7 (. m ≥ 15.75 . m ≤ 15.75 (orrect answeranswer-  ifficulty levellevel- $ 44. /ennifer wants to spend no more than 94++ 94++ on school clothes. clothes. Fhe spends 9;# on a Nacket and wants to buy some shirts that are on sale for 91+ each. Which inequality represents the number, s, of shirts /ennifer can buy A. s < 22 B. s > 30 C. s > 37 D. s < 38 (orrect answer- & ifficulty ifficult y level- 4 4. Teddy Teddy is delivering delivering boxes of paper paper to each each floor of an office office building. building. =ach box weighs #)  pounds, and Teddy Teddy himself weighs 1+ pounds. 8f the maximum capacity of an elevator is $,+++ pounds, or one ton, which of the following inequalities describes how many b boxes Teddy Teddy can safely take on each elevator trip without going over the capacity &. b ≥ 32 '. b ≤ 32 (. b ≥ 33

.

b ≤ 33

(orrect answeranswer-  ifficulty levellevel- 4 4#. & small airplane airplane can carry less than 1+#+ 1+#+ pounds of luggage and mail. Tuesday@s Tuesday@s load of mail weighs >+ pounds. 8f each passenger brings ;+ pounds of luggage, what is the greatest  possible number of passengers passengers that can travel travel on the airplane airplane on Tuesday Tuesday &. ; '.  (. 1 . 1# (orrect answer- & ifficulty ifficult y level- 4 4). Trevon Trevon is going going to buy a coat and a hat. The coat coat costs 4 times times as much as the hat. :e must must spend less than 9>. 8f the cost of the hat costs x dollars, which inequality best models this situation A. 3 x < 94 B. 3 x > 94 C. 4 x < 94 D. 4 x > 94 (orrect answeranswer- ( ifficulty level- 4 4;. & shippin shipping g compan company y charg charges es 94.) for the first first pound of a packag packagee and 91.#4 91.#4 for each each additional pound of the package. 8f Gobin wants to spend under 91++ on shipping, which inequality best represents how many pounds, p, Gobin can have in her particular package A.   3.46 + 1.53 p < 100 B.   3.46 + 1.53 ( p −1)< 100 C.   3.46 + 1.53 p − 1 < 100 D.   3.46 ( p −1)+ 1.53 < 100 (orrect answeranswer- ' ifficulty level- 4 4. 7embers of the wrestling wrestling team are planning to sell pre?ordered pre?ordered programs programs at matches. The The cost to print the programs is 91#+ plus 9+.#+ per program. They plan to sell each program for 9$. 8f profit is the amount of money earned from selling programs minus the expenses of printing the programs, how many programs must they sell to make a profit of at least 9#++ &. $)+ '. $)1 (. 44 . 4 (orrect answeranswer-  ifficulty levellevel-  4>. (hristoph (hristopher er has at most 91.#+ to spend at a convenience convenience store. store. :e buys one bag of bananas for 91.#+ and a bottle of apple Nuice for 91.##. 8f gasoline at this store costs 9$.4# per gallon, which of the following number of gallons can he buy for his car without exceeding his  budget &. .$4 '. ;.)1 (. 1+.$ . 1#.1; (orrect answer- & ifficulty ifficult y level- 

+. /onas is traveling traveling by bus to visit a friend friend who lives 4++ miles away. away. The friend friend has asked /onas to call at least 4+ minutes before arriving, so he can pick up /onas. /onas@s bus travels at a constant speed of # miles per hour. Which inequality shows the number of travel hours, t,  before which /onas /onas should call his friend friend 1 t≤6 &. 6  1 t≥6 '. 6 2 t≤6 (. 3 2 t≥6 . 3 (orrect answer- & ifficulty ifficult y level-  1. Ane particular particular hotel hotel in the downtown area area costs 9>+ a night. &n additiona additionall 1+C tax is added added to the original cost of the hotel room. There is also a one?time 91$ parking charge, and a family can expect to spend 94+ on tips during their stay. :ow many nights can a family spend at the hotel without exceeding their budget of 9)++ &. # '. ) (. )$ . )) (orrect answer- & ifficulty ifficult y level-  $. &n audiologist audiologist is testing a patient to determine determine the softest sound of a specific specific frequency frequency that the patient reports hearing. Fhe begins with a 1+ decibel !d'% sound for trial 1, then increases the volume by $d' for each trial after that. 8f the patient can hear all sounds of that frequency that have a volume louder than $)d', and t represents the trial numbers that the patient can hear, which of the following inequalities best models the situation described above &. 2 t + 10 ≥ 26 2 t + 10 > 26 '. (. 2 ( t −1 )+10 ≥ 26 . 2 ( t −1 )+10 > 26 (orrect answeranswer-  ifficulty levellevel-  4. & college college student student has equally weighted weighted exam exam grades of ;1, 4, and >+ on three 1++?poi 1++?point nt exams. The student has one more 1++?point exam that counts toward his grade and wants an exam average of at least . Which inequality represents the range of scores the student can get to achieve this desired exam average &. m≥ 8 '. m ≥ 79 (. m ≥ 81 . m ≥ 92 (orrect answeranswer-  ifficulty levellevel- 4 . (rys began began a workout workout regimen regimen at a local health club. club. Fhe began her workouts workouts at # minutes minutes each, but has since increased the workouts by # minutes per week. :er total workout time cannot exceed $ hours. 8f (rys has workouts for x weeks until she hits $ hours, which inequality best models the situation 45 x + 5 ≤ 120  A. 45 x + 5 ≥ 120  B.

C.  D.

45 + 5 ( x −1) ≤ 120 45 + 5 x ≤ 120

(orrect answeranswer- ( ifficulty level- 4 #. &nais &nais donates donates blood for for the first first time on 7ay 7ay 1#th, the 14#th day of the year. Fhe may donate again again every every #) days. days. 8f n is the number of total times which &nais donates donates this year, year, which equation best models the situation for n ≥ 1  56 n + 135 ≤ 365 &. '. 56 (n −1 )+135 ≤ 365 (. 56 n− 1 + 135 < 365 . 56 ( n + 135 ) < 365 (orrect answeranswer- ' ifficulty level- 4 ). /oanne /oanne and Gichard Gichard volunteer volunteer at a hospital. hospital. /oanne /oanne volunteers volunteers  hours more per week than Gichard does. 8n a given week, they do not volunteer for more than a combined total of 1) hours. 8f x is the number of hours that Gichard volunteers, which inequality best models this situation  A. x + 4 ≤ 16 2 x + 4 ≤ 16  B. C. 2 x + 8 ≤ 16 2 x −4 ≤ 16  D. (orrect answeranswer- ' ifficulty level-  ;. 7ikal 7ikal has a sum summer mer proNec proNectt in which which he must complet completee at least least 4# hours of community community service at a city park. =ach day that he goes to the park, he volunteers for ; hours. 8t takes him 1.# hours to get to the park each way, which also counts toward his community service hours. Which of the following inequalities can be used to find the number of days, d, 7ikal must volunteer at the park to complete his summer proNect 7 d > 35 &. 7 d ≥ 35 '. 7 d + 3 > 35 (. . 7 d + 3 d ≥ 35 (orrect answeranswer-  ifficulty levellevel-  . &n insurance agent sells an insurance insurance policy with with an out?of?pocket maximum of 9#,+++. Aut? Aut? of?pocket expenses include any deductibles and coinsurance that the client pays beyond the normal monthly premium. The client has a deductible of 9$#+. This means that the client has to pay all of the first 9$#+ of expenses insured by the policy. &fter that, the client has a $+C co?ins co?insura urance nce,, meani meaning ng the client client pays pays $+C of the insure insured d expens expenses es and the insura insurance nce compan company y pays pays the remain remainder der.. Which Which inequa inequality lity repre represe sents nts how much much the total total insure insured d expenses, x, could be if the client has not n ot yet reached the out?of?pocket maximum &. x < 26,000 '. x < 25,000 (. x < 24,000 . x < 23,750 (orrect answeranswer- ( ifficulty level- 