2.3E: Exercises - Solving Systems of Linear Inequalities in Two Variables

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

ⓐ $(5,−2)$
ⓑ $(−1,3)$

Answer: ⓐ false ⓑ true

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

ⓐ $(−4,−1)$
ⓑ $(8, 3)$

Answer: ⓐ false ⓑ true

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

ⓐ $(1, −3)$
ⓑ $(−4, 4)$

Answer: ⓐ false ⓑ true

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

4. $\left\{\begin{array} {l} y<−2x+2\\y\geq −x−1\end{array}\right.$

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.

5. $\left\{\begin{array} {l} y\geq −\frac{2}{3}x+2\\y>2x−3\end{array}\right.$

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.

6. $\left\{\begin{array} {l} x+2y<4\\y<x−2\end{array}\right.$

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.

7. $\left\{\begin{array} {l} x+4y\geq 8\\y\leq \frac{3}{4}x\end{array}\right.$

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.

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

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.

9. $\left\{\begin{array} {l} 2x+y>−6\\−x+2y\geq −4\end{array}\right.$

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.

10. $\left\{\begin{array} {l} x−3y>4\\y\leq −1\end{array}\right.$

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.

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

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.

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

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.

13. $\left\{\begin{array} {l} y\geq \frac{1}{2}x−1\\−2x+4y\geq 4\end{array}\right.$

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.

Solve Applications of Systems of Inequalities

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

14. Faran 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. Their van can hold at most 20 bags.

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

15. 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

16. 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

17. Marla is increasing her exercise routine by running and walking at least 4 miles each day. Her 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 she meet her goal by walking 3 miles and running 1 mile?
ⓓ Could she meet her goal by walking 2 miles and running 2 miles?

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

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