Problem 22 - Algebra 2 and Trigonometry

Question

In \(18-23,\) write and solve an equation or an inequality to solve the problem. Kim wants to buy an azalea plant for \(\$ 19\) and some delphinium plants for \(\$ 5\) each. She wants to spend less than \(\$ 49\) for the plants. At most how many delphinium plants can she buy?

Step-by-Step Solution

Answer

Kim can buy at most 5 delphinium plants.

Step 1: Understanding the Problem

The problem states that Kim wants to buy one azalea plant for \( \\( 19 \) and several delphinium plants for \( \\) 5 \) each. She has a budget of less than \( \$ 49 \). We need to find the maximum number of delphinium plants she can buy without exceeding her budget.

Step 2: Formulating the Inequality

Let \( x \) be the number of delphinium plants Kim wants to buy. The total cost for the azalea plant and \( x \) delphinium plants is \( 19 + 5x \). Since she wants to spend less than \( \$ 49 \), we can set up the inequality: \[ 19 + 5x < 49 \].

Step 3: Solving the Inequality

We solve for \( x \) to find out how many delphinium plants she can buy: 1. Subtract 19 from both sides: \[ 19 + 5x - 19 < 49 - 19 \] \[ 5x < 30 \]2. Divide both sides by 5: \[ x < \frac{30}{5} \] \[ x < 6 \].

Step 4: Conclusion

Since \( x \) must be an integer (as Kim can't buy a fraction of a plant), the maximum number of delphinium plants she can buy is 5. Buying 6 would exceed her budget. Therefore, Kim can buy at most 5 delphinium plants.

Key Concepts

Budget Constraints

Budget constraints are common in everyday life and help us prioritize spending. Imagine having a limited amount of money to spend on various items. You can't buy everything you want, so you need to make decisions on what to purchase within that limit. In the context of the problem, Kim's budget constraint means she wants to spend less than $49 on plants.
Understanding budget constraints allows you to determine how to optimize your spending without exceeding your total available funds.

• Identify the total amount you can spend. Here, it's less than $49.
• List the costs of each item you want to buy. Kim wants one azalea plant for $19 and delphinium plants for $5 each.
• Set up an inequality or an equation to determine how many items you can afford within your budget.

Budget constraints are pivotal in personal finance as they keep your spending in check and help you allocate resources effectively.

Linear Inequalities

Linear inequalities are like equations, but instead of equal signs, they use inequality signs such as < or >. They express a range of possible solutions, rather than a definite answer. This is useful when determining limits, like how many plants Kim can buy.For Kim, the inequality is:

• \(19 + 5x < 49\)

This shows that the cost of the azalea plant plus five times the cost of each delphinium plant must be less than $49. Solving this inequality helps find the maximum number of items she can purchase without overstepping her budget.
Linear inequalities are crucial in problem-solving because they help find solutions under specific conditions, ensuring limits are respected while achieving the desired outcome.

Problem Solving

Problem solving involves understanding the situation, formulating a plan, working through it, and concluding decisively. In the exercise, the first step was understanding that Kim wants plants without breaching her budget.
Once you grasp the problem's constraints and requirements:

• Formulate a clear strategy: Here, it's creating an inequality to reflect her budget constraints.
• Execute your plan: Solve the inequality \(19 + 5x < 49\) to find how many plants can be purchased.
• Arrive at a Conclusion: Decide on the number of items that meet the criteria without crossing the budget.

Embracing a structured problem-solving approach not only simplifies complex issues but makes solutions reliable and efficient.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They help represent real-life scenarios in mathematical form, making it easier to manipulate and solve problems.In this exercise, the expression \(19 + 5x\) represents the total cost of the plants Kim wants to buy:

• \(19\) is the fixed cost for the azalea plant.
• \(5x\) is the variable cost for the delphinium plants.

The beauty of algebraic expressions lies in their ability to succinctly summarize and link different aspects of a problem, allowing easier analysis and decision-making.
They form the backbone of setting up and solving equations and inequalities, translating real-world contexts into mathematical language, which enhances comprehension and facilitates solution-finding.