Problem 52 - Algebra 1: Concepts and Skills

Question

You have \(\$ 58\) and you want to buy a pair of jeans and a \(\$ 20\) T-shirt. There is a 6% sales tax. Let x represent the cost of the jeans. The following inequality models how much you can spend on the jeans. $$x+20+0.06(x+20) \leq 58$$ If the jeans cost $35, can you buy both the T-shirt and the jeans? Explain.

Step-by-Step Solution

Answer

No, you cannot afford to buy both the T-shirt and the jeans because the cost of the jeans, at \$35, exceeds the pre-calculated maximum allowable cost of \$34.72 while staying within the budget.

Step 1: Understand the inequality

The given inequality is \(x + 20 + 0.06*(x + 20) \leq 58\). The left side of the inequality represents the total cost of the jeans (x) and the t-shirt (\$20), plus the sales tax, represented as 6% of the total pre-tax cost (\(0.06*(x + 20)\)). The right side of the inequality (\$58) represents the total budget.

Step 2: Simplify the inequality

First, distribute the 0.06 through the parentheses to obtain the inequality: \(x + 20 + 0.06x + 1.2 \leq 58\). Then, combine like terms to simplify the inequality further to: \(1.06x + 21.2 \leq 58\).

Step 3: Solve the inequality for x

The objective is to isolate x. The first step is to subtract 21.2 from both sides, yielding the inequality: \(1.06x \leq 36.8\). Next, divide both sides by 1.06 to solve for x, yielding: \(x \leq 34.72\). This tells us that the cost of the jeans must be less than or equal to \$34.72 to stay within the budget.

Step 4: Answer the problem question

The jeans are given to cost \$35. This cost is higher than the maximum allowable cost of \$34.72 calculated in the previous step. Therefore, with a \$35 pair of jeans, you would not be able to buy both the t-shirt and the jeans within the budget.

Step 5: Formulate the conclusion

The conclusion based on the calculations is that the jeans, priced at \$35, would make the total cost exceed the available budget of \$58, even when the 6% sales tax is taken into account. So you cannot buy both the jeans and the t-shirt in this situation.

Key Concepts

Solving Inequalities

Inequalities in algebra are used to determine the range of possible values that a variable can take. Unlike equations, which find an exact value, inequalities provide a range or a limit. In this exercise, the inequality \(x + 20 + 0.06(x + 20) \leq 58\) was used to figure out the maximum cost of the jeans you can afford.
First, we needed to understand what each part of the inequality represents. The terms \(x\) and \(20\) denote the cost of the jeans and the T-shirt, respectively. \(0.06(x + 20)\) accounts for the 6% sales tax.
The goal when solving an inequality is to isolate the variable, here represented by \(x\). Once simplified, it becomes \(1.06x + 21.2 \leq 58\). Subtract \(21.2\) from both sides and then divide by \(1.06\) to solve for \(x\) revealing that \(x\) must be \(\leq 34.72\). This means the jeans can cost at most \$34.72 in order to keep the purchase within your budget.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations that are structured to form a function or a statement. When forming these expressions, it’s crucial to accurately translate real-world situations into mathematical terms.
In this exercise, the algebraic expression \(x + 20 + 0.06(x + 20)\) was modeled to represent your shopping scenario. Here, \(x\) is the variable that represents the unknown cost of the jeans. The number \(20\) is a constant representing the known cost of the T-shirt. The expression includes addition to combine these costs, and multiplication to apply the sales tax.
When working with algebraic expressions, it is important to:

• Identify what each variable represents.
• Accurately translate words into mathematical symbols.
• Combine like terms to simplify the expression.

Understanding these principles allows you to manipulate and solve the expression effectively, leading you to financial decisions like the one exemplified here.

Budgeting with Algebra

Budgeting with algebra involves creating mathematical models of budgeting constraints to ensure expenses don't exceed available funds. Algebraic inequalities are especially useful in this context as they define limits that mustn't be surpassed.
In cases like this exercise, you aim to purchase clothing within a specific budget. Your budgeting constraint was represented by the inequality \(x + 20 + 0.06(x + 20) \leq 58\). The purpose was to determine the maximum price you could pay for jeans.
To budget effectively using algebra:

• Clearly define each component of your expenses in mathematical terms.
• Consider all extras like taxes or fees and include them in your model.
• Perform operations that help you isolate variables representing unknown costs.

By following these guidelines, you can make informed purchasing decisions and determine if certain expenses fit within your financial constraints, exemplified here by not buying the jeans priced over the determined limit.