Practice Graphing Linear Inequalities

1. Noor wants to spend less than ‍$20 on condiments. Each kilogram of salt costs ‍$1.50, and each kilogram of pepper costs $2.50.

   Write an inequality that represents the number of kilograms of salt \((S)\) and pepper ‍\((P)\) Noor can buy on her budget.

2. Jocelyn enjoys building robots and painting pictures. It takes her a constant amount of time to build every robot, and it takes her a constant amount of time to paint every picture. She has 15 hours to build robots and paint pictures.

   In the following inequality, \(R\) represents the number of robots Jocelyn can build and \(P\) represents the number of paintings she can create over that time period:

   \(\qquad 2R+5P \leq 15\)
   

   Jocelyn wants to build 3 robots. How many paintings at most can she create in the time that remains?

   Choose 1 answer:

   1. Jocelyn can create at most 0 paintings.
   2. Jocelyn can create at most 1 painting.
   3. 
      
   4. Jocelyn can create at most 2 paintings.

   5. Jocelyn can create at most 3 paintings.

3. Goku wants to get over 9000 comments on his YouTube videos. He gets the same number of comments for each cat video that he uploads, and he gets the same number of comments for each dog video that he uploads.
   

   Let \(C\) represent the number of cat videos and \(D\) represent the number of dog videos that Goku should upload to achieve his goal.

   \(750C+450D > 9000\)
   

   According to the inequality, how many comments does each cat video get, and how many comments does each dog video get?

   Each cat video gets ___ comments, and each dog video gets ___ comments.

   Can Goku achieve his goal by uploading 8 cat videos and 7 dog videos?

   Choose 1 answer:

   1. Yes

   2. No

4. Miyoko's goal is to earn more than ‍$950 this week. She earns $250 for every day she works as a cryptographer, and earns $180 for every day she works as a geologist.
   

   Write an inequality that represents the number of days Miyoko should work as a cryptographer \(C\) and as a geologist ‍\(G\) to achieve her goal.
   

Source: Khan Academy, https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:inequalities-systems-graphs/x2f8bb11595b61c86:modeling-with-linear-inequalities/e/modeling-constraints
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1. We are given that each kilogram of salt costs ‍$1.50, and each kilogram of pepper costs $2.50.

   How can we express the total cost of the number of kilograms of salt and pepper Noor can buy on her budget?

   The cost of \(S\) kilograms of salt is ‍\(1.5S\), and the cost of ‍\(P\) kilograms of pepper is ‍\(2.5P\). Therefore, the total cost of buying some kilograms of salt and pepper is \(1.5S+2.5P\).

   We are also given that Noor wants to spend less than $20 on groceries. Let's use this condition to create the appropriate inequality.

   The following inequality represents the number of kilograms of salt and pepper Noor can buy on her budget:
   

   \(1.5S+2.5P<20\)

2. We are given that Jocelyn wants to build ‍3 robots. When we substitute ‍\(R=3\) in the given inequality, we obtain an inequality for ‍\(P\) alone:

   \(\qquad\begin{aligned}2R+5P &\leq 15\\\\
   2(3) + 5P &\leq 15 \\\\
   5P &\leq 9\\\\
   P&\leq1.8\end{aligned}\)
   

   So ‍\(P\) must be less than or equal to 1.8. However, we should remember that the number of paintings Jocelyn creates must be an integer.

   Since 1 is the biggest integer less than or equal to ‍1.8, we can conclude that Jocelyn can create at most 1 painting in the time that remains.

   Jocelyn can create at most 1 painting.

3. We are given the following inequality:

   \(750C+450D > 9000\)
   

   The coefficients of the variables \(C\) and \(D\) represent the number of comments each cat video and each dog video gets.

   Therefore, each cat video gets 750 comments, and each dog video gets 450 comments.

   Now let's check whether Goku can achieve his goal by uploading 8 cat videos and ‍7 dog videos. To do this, we substitute ‍\(C=8\) and \(D=7\) in the given inequality:

   \(\begin{aligned}750C+450D &> 9000\\\\
   750({8}) + 450({7}) &\stackrel{?}{>} 9000 \\\\
   9150 &\stackrel{\checkmark}> 9000\end{aligned}\)
   

   Since the inequality is true, Goku can achieve his goal by uploading 8 cat videos and 7 dog videos.

   In conclusion,

   • Each cat video gets 750 comments, and each dog video gets 450 comments. 

   • Yes, Goku can achieve his goal by uploading 8 cat videos and ‍7 dog videos.

   We are given that Miyoko earns ‍$250 for every day she works as a cryptographer and $180 for every day she works as a geologist.

   How can we express the total amount Miyoko can earn by working as a cryptographer for some days and as a geologist for some days?

   The amount earned by working \(C\) days as a cryptographer is ‍\(250C\), and the amount earned by working as a geologist for ‍\(G\) days is ‍\(180%\). Therefore, the total amount earned by working as a cryptographer and a geologist is \(250C+180G\).

   We are also given that Miyoko's goal is to earn more than $950 this week. Let's use this condition to create the appropriate inequality.

   The following inequality represents the number of days Miyoko should work as a cryptographer and as a geologist to achieve her goal:

   \(250C + 180G > 950\)