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

ⓐ $$(5,−2)$$
ⓑ $$(−1,3)$$

ⓐ false ⓑ true

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

ⓐ $$(−4,−1)$$
ⓑ $$(8, 3)$$

ⓐ false ⓑ true

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

ⓐ $$(1, −3)$$
ⓑ $$(−4, 4)$$

ⓐ false ⓑ true

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

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

The solution is the grey region.

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

The solution is the grey region.

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

The solution is the grey region.

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

The solution is the grey region.

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

The solution is the grey region.

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

The solution is the grey region.

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

The solution is the grey region.

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

The solution is the grey region.

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

No solution.

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

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 ⓐ $$\left\{\begin{array} {l} f\geq 0 \\ p\geq 0 \\ f+p\leq 202 \\ f+5p\leq 50\end{array}\right.$$ ⓒ 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 ⓐ $$\left\{\begin{array} {l} c\geq 0\\a\geq 0\\c+a\leq 24\\a\geq 3c\end{array}\right.$$ ⓒ 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?

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

ⓒ 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?

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