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Linear Inequalities in One Variable

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A shipping service charges a flat rate of 15.50toshipabox,plus15.50 to ship a box, plus 0.50 per ounce of the total weight of the box. A customer wants to ship a package and wants the total shipping cost to be at most $40.00. If the empty box weighs 14 ounces, what is the maximum weight, in ounces, of the contents that can be placed inside the box?

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Answer: 35

Answer

The maximum weight of the contents that can be placed inside the box is 35 ounces.
To find the maximum weight of the contents, let ww represent the weight of the contents in ounces. The total weight of the package is the sum of the content weight and the empty box weight, or w+14w + 14 ounces. The total shipping cost is the flat rate of 15.50plus15.50 plus 0.50 times the total weight, which is represented by the expression 15.50+0.50(w+14)15.50 + 0.50(w + 14). Since the customer wants the cost to be at most 40.00,wewritetheinequality40.00, we write the inequality 15.50 + 0.50(w + 14) \leq 40.00 .Distributing. Distributing 0.50 yields yields 15.50 + 0.50w + 7 \leq 40.00 .Combiningtheconstanttermsgives. Combining the constant terms gives 22.50 + 0.50w \leq 40.00 .Subtracting. Subtracting 22.50 frombothsidesresultsin from both sides results in 0.50w \leq 17.50 .Finally,dividingbothsidesby. Finally, dividing both sides by 0.50 gives gives w \leq 35$. Thus, the maximum weight of the contents is 35 ounces.

Step-by-Step Solution

1
Define the variable and write the inequality representing the total cost constraints.
15.50+0.50(w+14)≤40.0015.50 + 0.50(w + 14) \leq 40.00
The total weight is the sum of the contents (ww) and the empty box (14 ounces), and the cost is 15.50plus15.50 plus 0.50 per ounce, which cannot exceed $40.00.
2
Distribute the rate of $0.50 per ounce across the weight terms and simplify.
22.50+0.50w≤40.0022.50 + 0.50w \leq 40.00
Applying the distributive property gives 0.50w+7.000.50w + 7.00, and adding the flat rate 15.5015.50 simplifies the left side.
3
Isolate the variable term by subtracting 22.5022.50 from both sides.
0.50w≤17.500.50w \leq 17.50
This isolates the variable term 0.50w0.50w on one side of the inequality.
4
Solve for ww by dividing both sides by 0.500.50.
w≤35w \leq 35
Dividing by 0.500.50 (or multiplying by 2) yields the maximum weight limit for the contents.

Key Concept

Solving linear inequalities in one variable using distributive properties and combining like terms.
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