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You know that a system of linear equations is a set of two or more linear equations. These equations have the same variables, and the power on the independent variable, usually \(x\), is one.
Similarly, a system of linear inequalities is a set of two or more linear inequalities. These inequalities have the same variables, and the power on the independent variable, usually \(x\), is one. Note that in a system of inequalities, the linear inequalities can have the same or different inequality symbols. For example, this set is a system of linear inequalities.
\( y \geq \frac{1}{2}x + 3\)
\( y < 2x + 9 \).
Creating systems of linear inequalities from a written description is like creating a linear inequality. You start by defining your variables, then you choose the inequality symbols, and finally, you put it all together. For example:
Tia has a small business selling beach-themed jewelry. She has rented a booth to sell her jewelry at the Geocove waterpark. She sells bracelets for $4.00, and necklaces are $9.00 each. This weekend Tia needs to earn over $500. She also needs to sell no less than 75 pieces of jewelry.
Write a system of inequalities to represent this scenario.
The steps for writing a system of linear equalities from a description are shown in the table below. Click each step to see it applied to the example.
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Let \(x =\) the number of bracelets Tia sells. Let \(y =\) the number of necklaces Tia sells. |
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This system has two inequalities. You know Tia needs to earn “over” $500. You will use the \(>\) symbol for this one. You also know Tia needs to sell “no less than” 75 pieces of jewelry. Use the \(\geq\) symbol. |
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You have information about how much the jewelry costs and how much Tia needs to earn. The bracelets cost $4.00, and necklaces are $9.00 each. Tia needs to earn over $500. \(4x + 9y > 500\) Tia needs to sell no less than 75 pieces of jewelry. \(x + y \geq 75\) The system of inequalities is \( 4x + 9y > 500 \) \( x + y \geq 75 \) |
How well can you use a description to write a system of inequalities? Use the activity below to practice. Read the scenario on each tab. Then, use the steps shown in the example above to write a system of inequalities that represents the scenario. Be sure to check your answers.
Isla needs to restock sun hats and beach towels in the Geocove gift shop. Sun hats cost $5.50 each, and beach towels are $7.00 apiece. Isla can spend no more than $500 on her order and has space to store and display less than 50 total sun hats and beach towels.
Write a system of inequalities to represent this scenario.
\( 5.50s + 7.00b \leq 500 \)
\( s + b < 50 \)
If you need help arriving at this answer, click the Solution button.
| Step 1: Define your variables. |
Let \(s =\) the number of sun hats Let \(b =\) the number of beach towels |
| Step 2: Determine which inequality symbols to use. |
This system has two inequalities. Isla can spend no more than $500. Use the\(\ \leq\) symbol. Isla can display less than 50 total items. Use the \(<\) symbol. |
| Step 3: Write the system of inequalities. |
Write an inequality to represent the cost of the items. \(5.50s + 7.00b \leq 500\) Write an inequality to represent the number of items. \(s + b < 50\) Write the system of inequalities. \( s + b < 50 \) \( 5.50s + 7.00b \leq 500 \) |
Cate is buying plants and soil for her garden. Soil costs $3.99 per bag, and the plants cost $10 each. Cate needs to buy at least 5 plants and can spend up to $100 on these garden supplies.
Write a system of inequalities to represent this scenario.
\( 3.99b + 10p \leq 100 \)
\( p ≥ 5 \)
If you need help arriving at this answer, click the Solution button.
| Step 1: Define your variables. |
Let \(b\)\(=\) the number of bags of soil purchased. Let \(p =\) the number of plants purchased. |
| Step 2: Determine which inequality symbols to use. |
Cate can spend up to $100 on these garden supplies. Use the \(\leq\) symbol. She needs to buy at least 5 plants. Use the \(≥ \ \)symbol. |
| Step 3: Write the system of inequalities. |
Write an inequality to represent the cost of the soil and the plants. \(3.99b + 10p \leq 100\) Write an inequality to represent the number of plants Cate wants to buy. \(p ≥ 5\) Write the system of inequalities. \( 3.99b + 10p \leq 100 \) \( p ≥ 5 \) |
Yassin received a $50 gift card for an app store for his birthday. Games cost an average of $1.99 each, and songs are $0.99 each. Yassin wants to be able to purchase at least 18 items with his gift card and doesn’t want to spend more than the gift card is worth.
Write a system of inequalities to represent this scenario.
\( 1.99g + 0.99s \leq 50 \)
\( g + s \geq 18 \)
If you need help arriving at this answer, click the solution button.
| Step 1: Define your variables. |
Let \(g =\) the number of games Yassin can buy. Let \(s =\) the number of songs Yassin can buy. |
| Step 2: Determine which inequality symbols to use. |
Yassin does not want to spend more than the amount of his gift card. Use the symbol \(\leq\). Yassin wants to buy at least 18 items from the app store. Use the \(\geq\) symbol. |
| Step 3: Write the system of inequalities. |
Write an inequality to represent the cost of the games and songs. \(1.99g + 0.99s \leq 50\) Write an inequality to represent the number of items to buy. \(g + s \geq 18\) Write the system of inequalities. \( 1.99g + 0.99s \leq 50 \) \( g + s \geq 18 \) |
