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Mathematics LibreTexts

2.6E: Exercises

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    Practice Makes Perfect

    Graph Inequalities on the Number Line

    In the following exercises, graph each inequality on the number line and write in interval notation.

    1.  ⓐ  \(x<−2\)
        ⓑ  \(x\geq −3.5\)
        ⓒ  \(x\leq \frac{2}{3}\)

    2.  ⓐ  \(x>3\)
        ⓑ  \(x\leq −0.5\)
        ⓒ  \(x\geq \frac{1}{3}\)

    Answer

    The solution for x is greater than 3 on a number line has a left bracket 3 with shading to the right. The solution in interval notation is 3 to infinity within parentheses.


    The solution for x is less than or equal to negative 0.5 on a number line has a right bracket at negative 0.5 with shading to the left. The solution in interval notation is negative infinity to negative 0.5 within a parenthesis and a bracket.

    The solution for x is greater than or equal to one-third on a number line has a left bracket at one-third with shading to the right. The solution in interval notation is one-third to infinity within a bracket and a parenthesis.

    3.  ⓐ  \(x\geq −4\)
        ⓑ  \(x<2.5\)
        ⓒ  \(x>−\frac{3}{2}\)

    4.  ⓐ  \(x\leq 5\)
        ⓑ  \(x\geq −1.5\)
        ⓒ  \(x<−\frac{7}{3}\)

    Answer

    The solution for x is less than or equal to 5 on a number line has a right bracket with shading to the left. The solution in interval notation is negative infinity to 5 within a parenthesis and a bracket.


    The solution for x is greater than or equal to negative 1.5 on a number line has a left bracket with shading to the right. The solution in interval notation is negative 1.5 to infinity within a bracket and a parenthesis.


    The solution for x is less than negative seven-thirds on a number line has a right parenthesis with shading to the left. The solution in interval notation is negative infinity to negative seven-thirds within parentheses.

    5.  ⓐ  \(−5<x<2\)
        ⓑ  \(−3\leq x<−1\)
        ⓒ  \(0\leq x\leq 1.5\)

    6.  ⓐ  \(−2<x<0\)
        ⓑ  \(−5\leq x<−3\)
        ⓒ  \(0\leq x\leq 3.5\)

    Answer

    Negative 2 is less than x which is less than 0. There is an open circle at negative 2 and an open circle at 0 and shading between negative 2 and 0 on the number line. The interval notation is negative 2 and 0 within parentheses.


    Negative 5 is less than or equal to x which is less than negative 3. There is a closed circle at negative 5 and an open circle at negative 3 and shading between negative 5 and negative 3 on the number line. The interval notation is negative 5 and negative 3 within a bracket and a parenthesis.


    0 is less than or equal to x which is less than or equal to 3.5. There is a closed circle at 0 and a closed circle at 3.5 and shading between 0 and 3.5 on the number line. The interval notation is 0 and 3.5 within brackets.

    7.  ⓐ  \(−1<x<3\)
        ⓑ  \(−3<x\leq −2\)
        ⓒ  \(−1.25\leq x\leq 0\)

    8.  ⓐ  \(−4<x<2\)
        ⓑ  \(−5<x\leq −2\)
        ⓒ  \(−3.75\leq x\leq 0\)

    Answer

    Negative 4 is less than x which is less than 2. There is a open circle at negative 4 and an open circle at 2 and shading between negative 4 and 2 on the number line. The interval notation is negative 4 and 2 within parentheses.


    Negative 5 is less than x which is less than or equal to 2. There is a open circle at negative 5 and a closed circle at negative 2 and shading between negative 5 and negative 2 on the number line. The interval notation is negative 5 and negative 2 within a parenthesis and a bracket.


    Negative 3.75 is less than or equal to x which is less than or equal to 0. There is a closed circle at negative 3.75 and a closed circle at 0 and shading between negative 3.75 and 0 on the number line. The interval notation is negative 3.75 and 0 within brackets.

    Solve Linear Inequalities

    In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    9.  ⓐ  \(a+\frac{3}{4}\geq \frac{7}{10}\)
        ⓑ  \(8x>72\)
        ⓒ  \(20>\frac{2}{5}h\)

    10.  ⓐ  \(b+\frac{7}{8}\geq \frac{1}{6}\)
          ⓑ  \(6y<48\)
          ⓒ  \(40<\frac{5}{8}k\)

    Answer

    The solution is b is greater than or equal to negative seventeen twenty-fourths. The solution on a number line has a left bracket negative seventeen twenty-fourths with shading to the right. The solution in interval notation is negative seventeen twenty-fourths to infinity within a bracket and a parenthesis.


    The solution is y is less than 8. The solution on a number line has a right parenthesis at 8 with shading to the left. The solution in interval notation is negative infinity to 8 within parentheses.


    The solution is k is greater than 64. The solution on a number line has a left parenthesis at 64 with shading to the right. The solution in interval notation is 64 to infinity within parentheses.

    11.  ⓐ  \(f−\frac{13}{20}<−\frac{5}{12}\)
          ⓑ  \(9t\geq −27\)
          ⓒ  \(\frac{7}{6}j\geq 42\)

    12.  ⓐ  \(g−\frac{11}{12}<−\frac{5}{18}\)
          ⓑ  \(7s<−28\)
          ⓒ  \(\frac{9}{4}g\leq 36\)

    Answer

    The solution is g is less than twenty-three thirty-sixths. The solution on a number line has a right parenthesis at twenty-three thirty-sixths with shading to the left. The solution in interval notation is negative infinity to twenty-three thirty-sixths within parentheses.


    The solution is s is less than negative 4. The solution on a number line has a right parenthesis at negative 4 with shading to the left. The solution in interval notation is negative infinity to negative 4 within parentheses.


    The solution is g is less than or equal to 16. The solution on a number line has a right bracket at 16 with shading to the left. The solution in interval notation is negative infinity to 16 within parenthesis and a bracket.

    13.  ⓐ  \(−5u\geq 65\)
          ⓑ  \(\frac{a}{−3}\leq 9\)

    14.  ⓐ  \(−8v\leq 96\)
          ⓑ  \(\frac{b}{−10}\geq 30\)

    Answer

    The solution is v is greater than or equal to negative 12. The solution on a number line has a left bracket with shading to the right. The solution in interval notation is negative 12 to infinity within a bracket and a parenthesis.


    The solution is b is less than or equal to negative 300. The solution on a number line has a right bracket at negative 300 with shading to the left. The solution in interval notation is negative infinity to negative 300 within a parenthesis and a bracket.

    15.  ⓐ  \(−9c<126\)
          ⓑ  \(−25<\frac{p}{−5}\)

    16.  ⓐ  \(−7d>105\)
          ⓑ  \(−18>\frac{q}{−6}\)

    Answer

    The solution is d is less than negative 15. The solution on a number line has a right parentheses with shading to the left. The solution in interval notation is negative infinity to negative 15 within parentheses.


    The solution is q is greater than 108. The solution on a number line has a left parentheses at 108 with shading to the right. The solution in interval notation is 108 to infinity within parentheses.

    In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    17.  \(4v\geq 9v−40\)

    18.  \(5u\leq 8u−21\)

    Answer

    The solution is u is greater than or equal to 7. The solution on a number line has a left bracket at 7 with shading to the right. The solution in interval notation is 7 to infinity within a a bracket and a parenthesis.

    19.  \(13q<7q−29\)

    20.  \(9p>14p−18\)

    Answer

    The solution is p is less than eighteen fifths. The solution on a number line has a right parenthesis at eighteen fifths with shading to the left. The solution in interval notation negative infinity to eighteen fifths within parentheses.

    21.  \(12x+3(x+7)>10x−24\)

    22.  \(9y+5(y+3)<4y−35\)

    Answer

    The solution is y is less than negative 5. The solution on a number line has a right parenthesis at negative 5 with shading to the left. The solution in interval notation is negative infinity to negative 5 within parentheses.

    23.  \(6h−4(h−1)\leq 7h−11\)

    24.  \(4k−(k−2)\geq 7k−26\)

    Answer

    The solution is k is less than or equal to 7. The solution on a number line has a right bracket at 7with shading to the left. The solution in interval notation is negative infinity to 7 within a parenthesis and a bracket.

    25.  \(8m−2(14−m)\geq 7(m−4)+3m\)

    26.  \(6n−12(3−n)\leq 9(n−4)+9n\)

    Answer

    The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.

    27.  \(\frac{3}{4}b−\frac{1}{3}b<\frac{5}{12}b−\frac{1}{2}\)

    28.  \(9u+5(2u−5)\geq 12(u−1)+7u\)

    Answer

    The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.

    29.  \(\frac{2}{3}g−\frac{1}{2}(g−14)\leq \frac{1}{6}(g+42)\)

    30.  \(\frac{4}{5}h−\frac{2}{3}(h−9)\geq \frac{1}{15}(2h+90)\)

    Answer

    The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.

    31.  \(\frac{5}{6}a−\frac{1}{4}a>\frac{7}{12}a+\frac{2}{3}\)

    32.  \(12v+3(4v−1)\leq 19(v−2)+5v\)

    Answer

    The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.

    In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    33.  \(15k\leq −40\)

    34.  \(35k\geq −77\)

    Answer

    The solution is k is greater than or equal to negative eleven fifthss. The solution on a number line has a left bracket at negative eleven fifths with shading to the right. The solution in interval notation is negative eleven fifths to negative infinity within a bracket and a parenthesis.

    35.  \(23p−2(6−5p)>3(11p−4)\)

    36.  \(18q−4(10−3q)<5(6q−8)\)

    Answer

    The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.

    37.  \(−\frac{9}{4}x\geq −\frac{5}{12}\)

    38.  \(−\frac{21}{8}y\leq −\frac{15}{28}\)

    Answer

    The solution is y is greater than or equal to ten twenty-ninths. The solution on a number line has a left bracket at ten twenty-ninths with shading to the right. The solution in interval notation is ten twenty-ninths to infinity within a bracket and a parenthesis.

    39.  \(c+34<−99\)

    40.  \(d+29>−61\)

    Answer

    The solution is g is greater than negative 90. The solution on a number line has a left parenthesis at negative 90 with shading to the right. The solution in interval notation is negative 90 to infinity within parentheses.

    41.  \(\frac{m}{18}\geq −4\)

    42.  \(\frac{n}{13}\leq −6\)

    Answer

    The solution is n is less than or equal to negative 78. The solution on a number line has a right bracket at negative 78 with shading to the left. The solution in interval notation is negative infinity to negative 78 within a parenthesis and a bracket.

    Translate to an Inequality and Solve

    In the following exercises, translate and solve. Then graph the solution on the number line and write the solution in interval notation.

    43.  Three more than \(h\) is no less than \(25\).

    44.  Six more than \(k\) exceeds \(25\).

    Answer

    The inequality is k plus 6 is greater than 25. Its solution is k is greater than 19. The solution on a number line has a left parenthesis at 19 with shading to the right. The solution in interval notation is 19 to infinity within parentheses.

    45.  Ten less than \(w\) is at least \(39\).

    46.  Twelve less than \(x\) is no less than \(21\).

    Answer

    The inequality is x minus 12 is greater than or equal to 21. Its solution is x is greater than or equal to 33. The solution on a number line has a left bracket at 33 with shading to the right. The solution in interval notation is 33 to infinity within a bracket and a parenthesis.

    47.  Negative five times \(r\) is no more than \(95\).

    48.  Negative two times \(s\) is lower than \(56\).

    Answer

    The inequality is negative 2 s is less than 56. Its solution is s is greater than negative 28. The solution on a number line has a left parenthesis at negative 28 with shading to the right. The solution in interval notation is negative 28 to infinity within parentheses.

    49.  Nineteen less than \(b\) is at most \(−22\).

    50.  Fifteen less than \(a\) is at least \(−7\).

    Answer

    The inequality is a minus 15 is greater than or equal to negative 7. Its solution is a is greater than or equal to 8. The solution on a number line has a left bracket at 8 with shading to the right. The solution in interval notation is 8 to infinity within a bracket and a parenthesis.

    Solve Applications with Linear Inequalities

    In the following exercises, solve.

    51. Alan is loading a pallet with boxes that each weighs 45 pounds. The pallet can safely support no more than 900 pounds. How many boxes can he safely load onto the pallet?

    52. The elevator in Yehire’s apartment building has a sign that says the maximum weight is 2100 pounds. If the average weight of one person is 150 pounds, how many people can safely ride the elevator?

    Answer

    A maximum of 14 people can safely ride in the elevator.

    53. Andre is looking at apartments with three of his friends. They want the monthly rent to be no more than $2,360. If the roommates split the rent evenly among the four of them, what is the maximum rent each will pay?

    54. Arleen got a $20 gift card for the coffee shop. Her favorite iced drink costs $3.79. What is the maximum number of drinks she can buy with the gift card?

    Answer

    five drinks

    55. Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him $10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?

    56. Ryan charges his neighbors $17.50 to wash their car. How many cars must he wash next summer if his goal is to earn at least $1,500?

    Answer

    86 cars

    57. Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?

    58. Kimuyen needs to earn $4,150 per month in order to pay all her expenses. Her job pays her $3,475 per month plus 4% of her total sales. What is the minimum Kimuyen’s total sales must be in order for her to pay all her expenses?

    Answer

    $16,875

    59. Andre has been offered an entry-level job. The company offered him $48,000 per year plus 3.5% of his total sales. Andre knows that the average pay for this job is $62,000. What would Andre’s total sales need to be for his pay to be at least as high as the average pay for this job?

    60. Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?

    Answer

    $110,000

    61. Jake’s water bill is $24.80 per month plus $2.20 per ccf (hundred cubic feet) of water. What is the maximum number of ccf Jake can use if he wants his bill to be no more than $60?

    62. Kiyoshi’s phone plan costs $17.50 per month plus $0.15 per text message. What is the maximum number of text messages Kiyoshi can use so the phone bill is no more than $56.60?

    Answer

    260 messages

    63. Marlon’s TV plan costs $49.99 per month plus $5.49 per first-run movie. How many first-run movies can he watch if he wants to keep his monthly bill to be a maximum of $100?

    64. Kellen wants to rent a banquet room in a restaurant for her cousin’s baby shower. The restaurant charges $350 for the banquet room plus $32.50 per person for lunch. How many people can Kellen have at the shower if she wants the maximum cost to be $1,500?

    Answer

    35 people

    65. Moshde runs a hairstyling business from her house. She charges $45 for a haircut and style. Her monthly expenses are $960. She wants to be able to put at least $1,200 per month into her savings account order to open her own salon. How many “cut & styles” must she do to save at least $1,200 per month?

    66. Noe installs and configures software on home computers. He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?

    Answer

    32 jobs

    67. Katherine is a personal chef. She charges $115 per four-person meal. Her monthly expenses are $3,150. How many four-person meals must she sell in order to make a profit of at least $1,900?

    68. Melissa makes necklaces and sells them online. She charges $88 per necklace. Her monthly expenses are $3,745. How many necklaces must she sell if she wants to make a profit of at least $1,650?

    Answer

    62 necklaces

    69. Five student government officers want to go to the state convention. It will cost them $110 for registration, $375 for transportation and food, and $42 per person for the hotel. There is $450 budgeted for the convention in the student government savings account. They can earn the rest of the money they need by having a car wash. If they charge $5 per car, how many cars must they wash in order to have enough money to pay for the trip?

    70. Cesar is planning a four-day trip to visit his friend at a college in another state. It will cost him $198 for airfare, $56 for local transportation, and $45 per day for food. He has $189 in savings and can earn $35 for each lawn he mows. How many lawns must he mow to have enough money to pay for the trip?

    Answer

    seven lawns

    71. Alonzo works as a car detailer. He charges $175 per car. He is planning to move out of his parents’ house and rent his first apartment. He will need to pay $120 for application fees, $950 for security deposit, and first and last months’ rent at $1,140 per month. He has $1,810 in savings. How many cars must he detail to have enough money to rent the apartment?

    72. Eun-Kyung works as a tutor and earns $60 per hour. She has $792 in savings. She is planning an anniversary party for her parents. She would like to invite 40 guests. The party will cost her $1,520 for food and drinks and $150 for the photographer. She will also have a favor for each of the guests, and each favor will cost $7.50. How many hours must she tutor to have enough money for the party?

    Answer

    20 hours

    Everyday Math

    73. Maximum load on a stage In 2014, a high school stage collapsed in Fullerton, California, when 250 students got on stage for the finale of a musical production. Two dozen students were injured. The stage could support a maximum of 12,750 pounds. If the average weight of a student is assumed to be 140 pounds, what is the maximum number of students who could safely be on the stage?

    74. Maximum weight on a boat In 2004, a water taxi sank in Baltimore harbor and five people drowned. The water taxi had a maximum capacity of 3,500 pounds (25 people with average weight 140 pounds). The average weight of the 25 people on the water taxi when it sank was 168 pounds per person. What should the maximum number of people of this weight have been?

    Answer

    20 people

    75. Wedding budget Adele and Walter found the perfect venue for their wedding reception. The cost is $9850 for up to 100 guests, plus $38 for each additional guest. How many guests can attend if Adele and Walter want the total cost to be no more than $12,500?

    76. Shower budget Penny is planning a baby shower for her daughter-in-law. The restaurant charges $950 for up to 25 guests, plus $31.95 for each additional guest. How many guests can attend if Penny wants the total cost to be no more than $1,500?

    Answer

    42 guests

    Writing Exercises

    77. Explain why it is necessary to reverse the inequality when solving \(−5x>10\).

    78. Explain why it is necessary to reverse the inequality when solving \(n−3<12\).

    Answer

    Answers will vary.

    79. Find your last month’s phone bill and the hourly salary you are paid at your job. Calculate the number of hours of work it would take you to earn at least enough money to pay your phone bill by writing an appropriate inequality and then solving it. Do you feel this is an appropriate number of hours? Is this the appropriate phone plan for you?

    80. Find out how many units you have left, after this term, to achieve your college goal and estimate the number of units you can take each term in college. Calculate the number of terms it will take you to achieve your college goal by writing an appropriate inequality and then solving it. Is this an acceptable number of terms until you meet your goal? What are some ways you could accelerate this process?

    Answer

    Answers will vary.

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has four columns and five rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was graph inequalities on the number line. In row 3, the I can was solve linear inequalities. In row 4, the I can was translate words to an inequality and solve. In row 5, the I can was solve applications with linear inequalities.

    ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?