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Mathematics LibreTexts

4.8E: Exercises

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    Practice Makes Perfect

    Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

    In the following exercises, determine whether each ordered pair is a solution to the system.

    1. \(\left\{\begin{array} {l} 3x+y>5\\2x−y\leq 10\end{array}\right.\)

    ⓐ \((3,−3)\)
    ⓑ \((7,1)\)

    2. \(\left\{\begin{array} {l} 4x−y<10\\−2x+2y>−8\end{array}\right.\)

    ⓐ \((5,−2)\)
    ⓑ \((−1,3)\)

    Answer

    ⓐ false ⓑ true

    3. \(\left\{\begin{array} {l} y>\frac{2}{3}x−5\\x+\frac{1}{2}y\leq 4\end{array}\right.\)

    ⓐ \((6, −4)\)
    ⓑ \((3, 0)\)

    4. \(\left\{\begin{array} {l} y<\frac{3}{2}x+3\\ \frac{3}{4}x−2y<5\end{array}\right.\)

    ⓐ \((−4,−1)\)
    ⓑ \((8, 3)\)

    Answer

    ⓐ false ⓑ true

    5. \(\left\{\begin{array} {l} 7x+2y>14\\5x−y\leq 8\end{array}\right.\)

    ⓐ \((2, 3)\)
    ⓑ \((7, −1)\)

    6. \(\left\{\begin{array} {l} 6x−5y<20\\−2x+7y>−8 \end{array}\right.\)

    ⓐ \((1, −3)\)
    ⓑ \((−4, 4)\)

    Answer

    ⓐ false ⓑ true

    Solve a System of Linear Inequalities by Graphing

    In the following exercises, solve each system by graphing.

    7. \(\left\{\begin{array} {l} y\leq 3x+2\\y>x−1\end{array}\right.\)

    8. \(\left\{\begin{array} {l} y<−2x+2\\y\geq −x−1\end{array}\right.\)

    Answer

    The figure shows the graph of inequalities y less than minus two times x plus two and y greater than or equal to minus x minus one. Two intersecting lines are shown, one in red and the other in blue. The area bound by the two lines is shown in grey.

    The solution is the grey region.

    9. \(\left\{\begin{array} {l} y<2x−1\\y\leq −\frac{1}{2}x+4\end{array}\right.\)

    10. \(\left\{\begin{array} {l} y\geq −\frac{2}{3}x+2\\y>2x−3\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities y greater than or equal to minus two by three x plus two and y greater than two times x minus three. Two intersecting lines, one in red and the other in blue, are shown. The region bound by them is shown in grey.

    The solution is the grey region.

    11. \(\left\{\begin{array} {l} x−y>1\\y<−\frac{1}{4}x+3\end{array}\right.\)

    12. \(\left\{\begin{array} {l} x+2y<4\\y<x−2\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities x minus two times y less than four and y less than x minus two. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey.

    The solution is the grey region.

    13. \(\left\{\begin{array} {l} 3x−y\geq 6\\y\geq −\frac{1}{2}x\end{array}\right.\)

    14. \(\left\{\begin{array} {l} x+4y\geq 8\\y\leq \frac{3}{4}x\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities two times x plus four times y greater than or equal to eight and y less than or equal to minus three fourth of x. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

    The solution is the grey region.

    15. \(\left\{\begin{array} {l} 2x−5y<10\\3x+4y\geq 12\end{array}\right.\)

    16. \(\left\{\begin{array} {l} 3x−2y\leq 6\\−4x−2y>8\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities three times x minus two times y less than or equal to six and minus four times x minus two times y greater than eight. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

    The solution is the grey region.

    17. \(\left\{\begin{array} {l} 2x+2y>−4\\−x+3y\geq 9\end{array}\right.\)

    18. \(\left\{\begin{array} {l} 2x+y>−6\\−x+2y\geq −4\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities two times x plus y greater than minus six and minus x plus two times y greater than or equal to minus four. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

    The solution is the grey region.

    19. \(\left\{\begin{array} {l} x−2y<3\\y\leq 1\end{array}\right.\)

    20. \(\left\{\begin{array} {l} x−3y>4\\y\leq −1\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities x minus three times y greater than four and y less than or equal to minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

    The solution is the grey region.

    21. \(\left\{\begin{array} {l} y\geq −\frac{1}{2}x−3\\x\leq 2\end{array}\right.\)

    22. \(\left\{\begin{array} {l} y\leq −\frac{2}{3}x+5\\x\geq 3\end{array}\right.\)

    Answer

    The figure shows the graph of the inequality y less than or equal to minus two by three times x plus five and x greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

    The solution is the grey region.

    23. \(\left\{\begin{array} {l} y\geq \frac{3}{4}x−2\\y<2\end{array}\right.\)

    24. \(\left\{\begin{array} {l} y\leq −\frac{1}{2}x+3\\y<1\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities y less than or equal to minus half x plus three and y less than one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

    The solution is the grey region.

    25. \(\left\{\begin{array} {l} 3x−4y<8\\x<1\end{array}\right.\)

    26. \(\left\{\begin{array} {l} −3x+5y>10\\x>−1\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities minus three times x plus five times y greater than ten and x greater than minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

    The solution is the grey region.

    27. \(\left\{\begin{array} {l} x\geq 3\\y\leq 2\end{array}\right.\)

    28. \(\left\{\begin{array} {l} x\leq −1\\y\geq 3\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities x less than or equal to minus one and y greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

    The solution is the grey region.

    29. \(\left\{\begin{array} {l} 2x+4y>4 \\ y\leq −\frac{1}{2}x−2\end{array}\right.\)

    30. \(\left\{\begin{array} {l} x−3y\geq 6\\y>\frac{1}{3}x+1\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities x minus three times y greater than or equal to six and y greater than one third of x plus one. Two non intersecting lines, one in blue and the other in red, are shown.

    No solution.

    31. \(\left\{\begin{array} {l} −2x+6y<0\\6y>2x+4\end{array}\right.\)

    32. \(\left\{\begin{array} {l} −3x+6y>12\\4y\leq 2x−4\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities minus three times x plus six times y greater than twelve and four times y less than or equal to two times x minus four. Two non intersecting lines, one in blue and the other in red, are shown.

    No solution.

    33. \(\left\{\begin{array} {l} y\geq −3x+2\\3x+y>5\end{array}\right.\)

    34. \(\left\{\begin{array} {l} y\geq \frac{1}{2}x−1\\−2x+4y\geq 4\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities y greater than or equal to minus half x minus one and minus two times x plus four times y greater than or equal to four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.

    The solution is the grey region.

    35. \(\left\{\begin{array} {l} y\leq −\frac{1}{4}x−2\\x+4y<6\end{array}\right.\)

    36. \(\left\{\begin{array} {l} y\geq 3x−1\\−3x+y>−4\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities y greater than or equal to three times x minus one and minus three times x plus y greater than minus four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.

    The solution is the grey region.

    37. \(\left\{\begin{array} {l} 3y>x+2\\−2x+6y>8\end{array}\right.\)

    38. \(\left\{\begin{array} {l} y<\frac{3}{4}x−2\\−3x+4y<7\end{array}\right.\)

    Answer

    The figure shows the graph of the inequalities y less than three by fourth x minus two and minus three x plus four y less than seven. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.

    The solution is the grey region.

    Solve Applications of Systems of Inequalities

    In the following exercises, translate to a system of inequalities and solve.

    39. Caitlyn sells her drawings at the county fair. She wants to sell at least 60 drawings and has portraits and landscapes. She sells the portraits for $15 and the landscapes for $10. She needs to sell at least $800 worth of drawings in order to earn a profit.

    ⓐ Write a system of inequalities to model this situation.
    ⓑ Graph the system.
    ⓒ Will she make a profit if she sells 20 portraits and 35 landscapes?
    ⓓ Will she make a profit if she sells 50 portraits and 20 landscapes?

    40. Jake does not want to spend more than $50 on bags of fertilizer and peat moss for his garden. Fertilizer costs $2 a bag and peat moss costs $5 a bag. Jake’s van can hold at most 20 bags.

    ⓐ Write a system of inequalities to model this situation.
    ⓑ Graph the system.
    ⓒ Can he buy 15 bags of fertilizer and 4 bags of peat moss?
    ⓓ Can he buy 10 bags of fertilizer and 10 bags of peat moss?

    Answer

    ⓐ \(\left\{\begin{array} {l} f\geq 0 \\ p\geq 0 \\ f+p\leq 202 \\ f+5p\leq 50\end{array}\right.\)

    The figure shows the graph of the inequalities f plus p less than or equal to twenty and two f and five p less than or equal to fifty. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

    ⓒ yes
    ⓓ no

    41. Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is $7.

    ⓐ Write a system of inequalities to model this situation.
    ⓑ Graph the system.
    ⓒ Can she mail 60 cards and 26 packages?
    ⓓ Can she mail 90 cards and 40 packages?

    42. Juan is studying for his final exams in chemistry and algebra. he knows he only has 24 hours to study, and it will take him at least three times as long to study for algebra than chemistry.

    ⓐ Write a system of inequalities to model this situation.
    ⓑ Graph the system.
    ⓒ Can he spend 4 hours on chemistry and 20 hours on algebra?
    ⓓ Can he spend 6 hours on chemistry and 18 hours on algebra?

    Answer

    ⓐ \(\left\{\begin{array} {l} c\geq 0\\a\geq 0\\c+a\leq 24\\a\geq 3c\end{array}\right.\)

    The figure shows the graph of the inequalities c plus a less than or equal to twenty four and a greater than or equal to three times c. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

    ⓒ yes
    ⓓ no

    43. Jocelyn is pregnant and so she needs to eat at least 500 more calories a day than usual. When buying groceries one day with a budget of $15 for the extra food, she buys bananas that have 90 calories each and chocolate granola bars that have 150 calories each. The bananas cost $0.35 each and the granola bars cost $2.50 each.

    ⓐ Write a system of inequalities to model this situation.
    ⓑ Graph the system.
    ⓒ Could she buy 5 bananas and 6 granola bars?
    ⓓ Could she buy 3 bananas and 4 granola bars?

    44. Mark is attempting to build muscle mass and so he needs to eat at least an additional 80 grams of protein a day. A bottle of protein water costs $3.20 and a protein bar costs $1.75. The protein water supplies 27 grams of protein and the bar supplies 16 gram. If he has $10 dollars to spend

    ⓐ Write a system of inequalities to model this situation.
    ⓑ Graph the system.
    ⓒ Could he buy 3 bottles of protein water and 1 protein bar?
    ⓓ Could he buy no bottles of protein water and 5 protein bars?

    Answer

    ⓐ \(\left\{\begin{array} {l} w\geq 0\\b\geq 0\\27w+16b>80\\3.20w+1.75b\leq 10\end{array}\right.\)

    The figure shows the graph of the inequalities twenty seven times w plus sixteen times b greater than eighty and three point two times w plus one point seven five b less than or equal to ten. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

    ⓒ no
    ⓓ yes

    45. Jocelyn desires to increase both her protein consumption and caloric intake. She desires to have at least 35 more grams of protein each day and no more than an additional 200 calories daily. An ounce of cheddar cheese has 7 grams of protein and 110 calories. An ounce of parmesan cheese has 11 grams of protein and 22 calories.

    ⓐ Write a system of inequalities to model this situation.
    ⓑ Graph the system.
    ⓒ Could she eat 1 ounce of cheddar cheese and 3 ounces of parmesan cheese?
    ⓓ Could she eat 2 ounces of cheddar cheese and 1 ounce of parmesan cheese?

    46. Mark is increasing his exercise routine by running and walking at least 4 miles each day. His goal is to burn a minimum of 1500 calories from this exercise. Walking burns 270 calories/mile and running burns 650 calories.

    ⓐ Write a system of inequalities to model this situation.
    ⓑ Graph the system.
    ⓒ Could he meet his goal by walking 3 miles and running 1 mile?
    ⓓ Could he his goal by walking 2 miles and running 2 mile

    Answer

    ⓐ \(\left\{\begin{array} {l} w\geq 0\\r\geq 0\\w+r\geq 4\\270w+650r\geq 1500\end{array}\right.\)

    The figure shows the graph of the inequalities w plus r greater than or equals to four and two seventy w plus six fifty r greater than or equal to fifteen hundred. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

    ⓒ no
    ⓓ yes

    Writing Exercises

    47. Graph the inequality \(x−y\geq 3\). How do you know which side of the line \(x−y=3\) should be shaded?

    48. Graph the system \(\left\{\begin{array} {l} x+2y\leq 6 \\ y\geq −\frac{1}{2}x−4\end{array}\right.\). What does the solution mean?

    Answer

    Answers will vary.

    Self Check

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

    The figure shows a table with four columns and four rows. The first row is the title row. The four titles are I can…, confidently, with some help and No – I don’t get it! In the second row of the first column, the text says ‘determine whether an ordered pair is a solution of a system of linear inequalities.’ In the third row of the first column, the text says ‘solve applications of systems of inequalities.’

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


    4.8E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.