Problem 71 - Algebra 1

Question

BUYING JEANS You have 58 dollar, and you want to buy a pair of jeans and a \(\$ 20\) T-shirt. There is a \(6 \%\) sales tax. If \(x\) represents the cost of the jeans, then the following inequality is a model that shows 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?

Step-by-Step Solution

Answer

No, you cannot afford to buy both the jeans priced at $35 and the T-shirt priced at $20 with $58 considering a 6% sales tax.

Step 1: Substitute the cost of the jeans

Start by substituting the cost of the jeans (\(x = 35\)) into the inequality: \[35+20+0.06(35+20) \leq 58\]

Step 2: Simplify the equation

Simplify and calculate the left side of the inequality first:\[0.06(35+20) = 3.3\]Add up all the numbers:\[35 + 20 + 3.3 = 58.3\]The inequality becomes:\[58.3 \leq 58\]

Step 3: Evaluate the inequality

Evaluate the inequality:Since 58.3 is greater than 58, the inequality is not satisfied. Therefore, with $58 you cannot afford to buy both the jeans (costing $35) and the T-shirt (costing $20), taking into account a 6% sales tax.

Key Concepts

Algebraic Expressions

When working with mathematical problems, it’s essential to understand algebraic expressions. These are combinations of numbers, variables (like the letter x), and arithmetic operations like addition, subtraction, multiplication, and division. The expression represents a value and is a cornerstone in algebra. In the context of the exercise, the expression x + 20 + 0.06(x + 20) encapsulates the total cost of purchasing jeans and a T-shirt, including sales tax.

The variable x stands for an unknown quantity, in this case, the cost of the jeans, which allows us to model and solve practical situations, like determining if you have enough money to make a purchase. Understanding how to work with these expressions, including substituting values and simplifying, is crucial when trying to find solutions to inequalities and equations. Think of algebraic expressions as a way to create a numeric model of real-world scenarios, which can then be analyzed and solved.

Inequality Models

Inequalities are like equations, but instead of an equal sign, they have symbols like <, >, ≤, or ≥ that show a relationship where one side is less than, greater than, or equal to the other side up to a certain point. When we talk about inequality models, we are taking situations from the real world and showing the relationship between quantities using these symbols.

In this textbook exercise, you have modeled your budget constraint with an inequality. The inequality x + 20 + 0.06(x + 20) ≤ 58 showcases that the cost of the jeans plus the T-shirt, including the sales tax, must be less than or equal to your available funds of $58. To solve this, you substitute known values and carry out arithmetic operations to see if the inequality holds true. In this case, when determining if you can afford both items at a given price, we found that the inequality does not hold, which means the total cost exceeds your budget.

Sales Tax Calculation

Sales tax is an additional amount of money, calculated as a percentage of the purchase price of goods and services, that consumers are required to pay to a governing body. To calculate sales tax, you simply multiply the tax rate (as a decimal) by the total purchase amount before tax. This is demonstrated in the exercise with a 6% sales tax rate applied to the cost of jeans and a T-shirt. The calculation is shown as 0.06(x + 20).

Remember, it’s essential to convert the percentage to a decimal by dividing by 100 before multiplying. For instance, 6% becomes 0.06. Once you have the sales tax amount, add it to the initial cost to find the total. The ability to accurately calculate sales tax is not only useful in algebra but is a practical life skill for managing personal finances and understanding the true cost of purchases.