Scenario: Planning a Party
You are planning a birthday party and need to buy some supplies. You have a budget of $150 to spend on decorations and snacks.
- You decide that you want to spend no more than $70 on decorations.
- You find a package of decorations that costs $30 and additional decorations that cost $10 each.
- You also want to buy snacks, which are going to cost you $5 per person attending the party.
You expect around 10 people, but you want to make sure you have enough money left over for at least half of the expected guests.
Creating the Inequality and Equation:
Let \( x \) represent the number of additional decoration packages you plan to buy.
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Inequality for Decorations:
The total cost of decorations must be less than or equal to $70.
The cost can be represented as:
\[ 30 + 10x \leq 70 \] -
Equation for Snacks:
You want to ensure that you have money left over for snacks for at least half the attendees. Since you expect 10 people, you will want to budget for at least 5 people.
Your remaining budget after decorations is $150 - (30 + 10x) for snacks. The cost for snacks is $5 per person.
Thus, we can set up the equation:
\[ 150 - (30 + 10x) = 5 \cdot 5 \]
Solving the Inequality and Equation:
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Solving the Inequality: \[ 30 + 10x \leq 70 \] Subtract 30 from both sides: \[ 10x \leq 40 \] Divide by 10: \[ x \leq 4 \]
This means you can buy up to 4 additional decoration packages.
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Solving the Equation for Snacks: \[ 150 - (30 + 10x) = 25 \] Expand the equation: \[ 150 - 30 - 10x = 25 \] Combine like terms: \[ 120 - 10x = 25 \] Subtract 120 from both sides: \[ -10x = -95 \] Divide by -10: \[ x = 9.5 \]
Since the number of extra packages must be a whole number, you would round down to 9. However, based on your inequality and budget constraints, you can only buy 4 additional decoration packages.
This scenario showcases how inequalities and equations can be applied to practical budgeting problems.
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