Antoine has $18.20 to spend on some oranges and a pumpkin. Oranges cost $1.30 per pound, and a pumpkin costs $5.20.
The inequality [tex]$1.30 x+5.20 \leq 18.20$[/tex] models this situation, where [tex]$x$[/tex] is the number of pounds of oranges.
Solve the inequality. How many pounds of oranges can Antoine buy?
A. [tex]$x \leq 10$[/tex]; Antoine can buy 10 pounds or less of oranges.
B. [tex]$x \leq 8$[/tex]; Antoine can buy 8 pounds or less of oranges.
C. [tex]$x \geq 8$[/tex]; Antoine can buy 8 pounds or more of oranges.
D. [tex]$x \geq 10$[/tex]; Antoine can buy 10 pounds or more of oranges.
Answer (2)
Subtract 5.20 from both sides of the inequality: 1.30 x ≤ 18.20 − 5.20 .
Simplify the right side: 1.30 x ≤ 13.00 .
Divide both sides by 1.30 : x ≤ 1.30 13.00 .
Calculate the value: x ≤ 10 . Therefore, Antoine can buy 10 pounds or less of oranges: x ≤ 10 .
Explanation
Understanding the Problem Let's analyze the problem. Antoine has a budget of $18.20 and wants to buy oranges and a pumpkin. The price of oranges is $1.30 per pound, and the pumpkin costs $5.20 . The inequality 1.30 x + 5.20 ≤ 18.20 represents this situation, where x is the number of pounds of oranges Antoine can buy. Our goal is to solve this inequality to find the maximum number of pounds of oranges Antoine can purchase.
Isolating the x term First, we need to isolate the term with x . To do this, we subtract 5.20 from both sides of the inequality:
1.30 x + 5.20 − 5.20 ≤ 18.20 − 5.20
Simplifying the inequality Now, simplify the inequality:
1.30 x ≤ 13.00
Solving for x Next, we need to solve for x by dividing both sides of the inequality by 1.30 :
1.30 1.30 x ≤ 1.30 13.00
Calculating the upper bound for x Now, perform the division:
x ≤ 10
Final Answer The solution to the inequality is x ≤ 10 . This means Antoine can buy 10 pounds or less of oranges. Therefore, the correct answer is:
x ≤ 10 ; Antoine can buy 10 pounds or less of oranges.
Examples
Imagine you're planning a picnic and have a limited budget. You need to buy ingredients like fruits and snacks. This problem is similar to figuring out how many pounds of fruit you can buy while staying within your budget, considering the cost of other essential items like drinks or sandwiches. By solving the inequality, you ensure you don't overspend and have a delightful picnic without breaking the bank. This type of problem helps in everyday budgeting and resource allocation.
Antoine can spend a maximum of $18.20 on oranges and a pumpkin, leading to the inequality 1.30 x + 5.20 ≤ 18.20 . After solving, we find that x ≤ 10 , meaning he can buy 10 pounds or less of oranges. Therefore, the correct choice is A.
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