In this explainer, we will learn how to solve applications of systems of inequalities by translating each condition into an inequality.
A system of linear inequalities (represented by , and ) is a set of two or more linear inequalities in several variables. It is used when a problem requires a range of solutions and there is more than one constraint on those solutions; for example, a shop trying to buy stock with a given budget.
Recall that, in a graph representing a system of inequalities, shading above or to the right means greater than, while shading below or to the left means less than a particular line defined by , , or the general line . In addition, we should also take the boundary of the region into account, where a solid line means equal to, while a dashed line means not equal to.
The intersection of the regions of each of the inequalities in a system is where the set of solutions lie, as this region satisfies every inequality in the system. We only include the values at the edges of the intersections of the region if there is a solid line on both, as all inequalities need to be satisfied and a strict inequality, represented by a dashed line, excludes it from the solution set.
For example, consider the system of inequalities
The region where all the shaded regions intersect is where the systems of inequalities are satisfied.
Letβs consider a real-world example where we translate conditions into a system of inequalities. As a one-variable example, suppose that a theme park has 3 rides that someone wants to try. The swing ride requires riders to be at least 1 m tall, the roller coaster requires riders to be at least 1.3 m tall, and the carousel requires riders to be between 0.8 m and 1.4 m tall.
If we denote the height of the rider as , then the constraint from the swing ride, at least 1 m tall, can be translated as
Similarly, the constraint for the roller coaster, at least 1.3 m tall, can be translated as
The last constraint, that the carousel requires riders to be between 0.8 m and 1.4 m, can be translated as
Thus, the system of inequalities representing this situation is
We can also combine these inequalities. First, note that we can ignore the first constraint, , since the second constraint, , takes this into account. Thus, we have the ineuqalities
Now, the second inequality is equivalent to and , and we can similarly combine the former with to obtain the inequalities and , which is equivalent to
Now, letβs consider another real-world example with linear inequalities with more than one variable. Suppose we want to make a rectangular garden and we have a certain amount of wire fencing, up to 40 metres, to place around the perimeter.
How can we represent this situation as a system of inequalities?
Firstly, let be the length in metres and be the width in metres, giving us the following rectangle:
Since the length and the width cannot be negative numbers, this translates into the condition
Note that the perimeter of the rectangle is , and since this cannot exceed 40 metres, this means that , which translates into the condition
To summarize, the system of inequalities representing this situation would be
This system can be visually represented in the first quadrant as follows:
The values of and that satisfy all the inequalities lie inside the intersection of the three shaded regions.
Now, letβs consider a few examples where we obtain a system of inequalities from a word problem. In the first example, we will state the system of inequalities for a shepherd wanting to build a rectangular sheep barn.
Example 1: Stating the System of Inequalities That Describes a Given Situation
A shepherd wants to build a rectangular sheep barn. The length of the barn must be more than 88 m and its perimeter must be less than 253 m. Derive the system of inequalities that describes the situation, denoting the length of the barn by and its width by .
Answer
In this example, we will state the systems of inequalities that satisfy the conditions for the shepherd to build a rectangular sheep barn.
Since the length of the barn must be more than 88 m and the width of the barn cannot be a negative value, we have the conditions
The perimeter of a polygon is the sum of its side lengths. Since the barn is rectangular, its perimeter, , is given by , and since this must be less than 253 m, we have
To summarize, the system of inequalities for each condition for the given situation is
Now, letβs determine the system of inequalities from a situation describing two types of nails a carpenter wants to buy.
Example 2: Determining the System of Inequalities That Describes a Given Situation
A carpenter wants to buy two types of nails; the first type costs 6 pounds per kilogram, and the second type costs 9 pounds per kilogram. He needs at least 5 kg of the first type and at least 7 kg of the second. He can spend less than 55 pounds. Using to represent the amount of the first type and to represent the second type, state the system of inequalities that represents this situation.
Answer
In this example, we will state the systems of inequalities that satisfy the conditions for a carpenter who wants to purchase two types of nails.
Since and are the amounts of nails (in kilograms) from the first and second type, respectively, and the carpenter needs at least 5 kg of the first type and 7 kg of the second, we have the condition
As the first type costs 6 pounds per kilogram and the second type costs 9 pounds per kilogram, the total price for each type would be and respectively. The sum of these has to be less than 55 pounds, and thus we have
To summarize, the system of inequalities for each condition for the given situation is
We could also have multiple linear inequalities (i.e., different regions defined above or below multiple straight lines), depending on the situation. Letβs consider another real-world example for this. Suppose we want to build a wardrobe and purchase two items, wooden planks and a pack of nails, and we want the number of wooden planks to be at least 7. We also know that the wooden planks cost $5 each and the pack of nails cost $3 and we have $75 in total to spend, so the final amount should be at least this. For each wooden plank, we should also have at least 4 nails, knowing that each pack contains 8 nails.
How do we represent this situation as a system of inequalities? This system can be represented as follows.
Firstly, letβs denote the number of wooden planks as and the number of packs of nails as . Since the number of wooden planks and packs of nails cannot be a negative number, this translates into the condition
Since we require that the number of wooden planks to be at least 7, this translates into the condition
This means we can ignore the condition , since this is already taken into account, because if , then it also means that . Now, since the wooden planks cost $5 each, the total amount spent on the wooden planks will be
Similarly, the packs of nails cost $3 each, so the total amount spent on the packs of nails will be
The total amount spent will be the sum of these, , and we have at most $75 to spend, which translates to
Since we are told that each wooden plank should have at least 4 nails, the total number of nails will be greater than or equal to 4 times the total number of planks, knowing that each pack contains 8 nails:
To summarize, the system of inequalities representing this situation will be
We can also represent this system visually as follows:
The values of and that satisfy all the inequalities lie inside the intersection of the four shaded regions.
Now, letβs look at a few examples to practice and deepen our understanding of the applications of systems of inequalities by translating each condition into an inequality. For the first example, we will determine the system of inequalities for a student taking a test.
Example 3: Determining the System of Inequalities That Describes a Given Situation
A teacher gave his students 100 minutes to solve a test that has two sections: section A and section B. The students had to answer at least 4 questions from section A and at least 6 questions from section B and answer at least 11 questions in total. If a girl answered each question in section A in 3 minutes and each question in section B in 6 minutes, derive the system of inequalities that would help to know how many questions she tried to solve in each section. Use to represent the number of questions answered from section A and to represent the number from section B.
Answer
In this example, we will state the systems of inequalities that satisfy the conditions for the student to know how many questions she tried to solve in a particular test.
Since and are the numbers of questions answered from section A and section B and the students had to answer at least 4 and 6, respectively, we have
As they also had to answer at least 11 questions in total, which is the sum of the numbers of questions answered from section A, , and the number of questions answered from section B, , we have
Since the girl answered each question from section A in 3 minutes and each from section B in 6 minutes, the total number of minutes for section A and B would be and respectively. The sum of these has to be at most 100 minutes, which is the maximum time given to solve the test; thus, we have
To summarize, the system of inequalities for each condition for the given situation is
In the next example, we will state the system of inequalities for a person buying two different types of candles.
Example 4: Stating the System of Inequalities That Describes a Given Situation
Shady is going to the store to buy candles. Small candles cost $3 and large candles cost $5. He needs to buy at least 20 candles, and he cannot spend more than $80. Write a system of linear inequalities that represents the situation, using to represent the number of small candles and to represent the number of large candles.
Answer
In this example, we will state the systems of inequalities that satisfy the conditions for Shady wanting to purchase two different types of candles, small and large.
Since and are the numbers of small and large candles, respectively, and these have to be nonnegative, we have
As Shady needs to buy at least 20 candles, the sum of the numbers of small and large candles, and , has to be greater than or equal to 20, and we have the condition
Since the small candles cost $3 and the large cost $5, the total for each will be and respectively. The sum of this will be the total cost of the candles, which cannot be more than 80; thus,
To summarize, the system of inequalities for each condition for the given situation is
Now, letβs consider an example where we determine the system of inequalities for a baby food factory that produces two types of baby food.
Example 5: Determining the System of Inequalities That Describes a Given Situation
A baby food factory produces two types of baby food. The first type contains 2 units of vitamin (A) and 3 units of vitamin (B) per gram. The second type contains 3 units of vitamin (A) and 2 units of vitamin (B) per gram. If a baby needs at least 100 units of vitamin (A) and 120 units of vitamin (B) per day, state the system of inequalities that describes the food that the baby must eat each day to meet these requirements. Use to represent the mass of the first type of baby food (in grams) and to represent the mass of the second type of baby food (in grams).
Answer
In this example, we will state the systems of inequalities that satisfy the conditions for the amount a baby must eat each day to meet the requirement on vitamin (A) and vitamin (B).
Since and are the masses of the first and second types of baby food (in grams), respectively, and these have to be nonnegative, we have
Since the first type contains 2 units of vitamin (A) and the second type contains 3 units of vitamin (A) per gram, the total number of vitamin (A) for each type is and respectively. The sum of these has to be at least 100 units, as the baby needs at least this many units, and thus
Similarly, the first type contains 3 units of vitamin (B) and the second type contains 2 units of vitamin (B) per gram, and the total number of units of vitamin (B) for each type is and respectively. The sum of these has to be at least 120 units, and thus
To summarize, the system of inequalities for each condition for the given situation is
Finally, letβs consider an example where we determine the system of inequalities for a candy manufacturer supplying a combination of different types of cookies to a speficic bakery.
Example 6: Determining the System of Inequalities That Describes a Given Situation
A candy manufacturer has 30 kg of chocolate cookies and 60 kg of vanilla cookies. Sales will be made in two different combinations. The first combination will be one-quarter chocolate cookies and three-quarters vanilla cookies by weight, while the second combination will be half chocolate cookies and half vanilla cookies by weight. There is a contract requiring that at least 20 kg of the second combination should be supplied to a specific bakery.
Which of the following systems of inequalities represents the number of kilograms of the first and second combinations that will be sold?
Let be the number of kilograms of the first combination and the number of kilograms of the second combination.
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,
Answer
In this example, we will state the systems of inequalities that satisfy the conditions for the number of kilograms of two different types of cookies to be sold.
Since and are the numbers of kilograms in the first and second combination, respectively, and these have to be nonnegative, and the contract requires at least 20 kg of the second combination, we have
As the first combination requires that one-quarter of the weight be from chocolate cookies and the second combination requires that half of the weight be from chocolate cookies, the total weight of chocolate cookies for each combination is and respectively. The sum of these is no greater than 30 kg, the maximum weight of the chocolate cookies, which gives us
We can also multiply this by 4 to write it as
Similarly, as the first combination requires three-quarters of the weight to be vanilla cookies and the second combination requires half of the weight to be vanilla cookies, the total amount of vanilla cookies for each combination is and respectively. The sum of these should be no greater than 60 kg, the maximum weight of vanilla cookies; thus, we have
We can also multiply this by 4 to write it as
To summarize, the system of inequalities for each condition for the given situation is
This is option A.
Key Points
- In a given situation, in order to state the system of inequalities, we should label each of the quantities or .
- If the quantities, and , are values that can never be negative, such as the length or width, then we should always start with
- We may be given other constraints for the quantities and , such as a minimum and/or maximum value for each.
- Additional linear inequalities of the form can be translated from constraints given for the total combination of quantities, taking into account the weighting of each quantity, such as the price of each quantity and the maximum expenditure.
