Problem 54 - Intermediate Algebra

Question

Gardening. During an Arbor Day sale, a garden store sold more than \(\$ 2,000\) worth of 6 -foot maple and 5 -foot pine trees. A 6-foot maple sold for \(\$ 100\) and a 5 -foot pine sold for S125. Write an inequality that describes the possible combinations of the number of the maple trees \((x)\) and the number of pine trees \((y)\) that could have been sold during the sale. Graph the inequality and give three ordered-pair solutions.

Step-by-Step Solution

Answer

The inequality is \(4x + 5y > 80\), and example solutions are (10, 12), (15, 8), (5, 14).

Step 1: Identify Variables and Costs

In this problem, we are dealing with two types of trees: maple trees, which we will represent with the variable \(x\), and pine trees, represented by \(y\). Each 6-foot maple tree is sold for \(\\(100\), and each 5-foot pine tree is sold for \(\\)125\). The total sales amount must exceed \(\$2000\).

Step 2: Set Up the Inequality

Since each maple costs \(\\(100\) and each pine costs \(\\)125\), the total revenue can be expressed as \(100x + 125y\). The problem states that the revenue is more than \(\$2000\), so our inequality is:\[100x + 125y > 2000\]

Step 3: Simplifying the Inequality

To make the inequality easier to work with, let's divide through by 25 to simplify:\[\frac{100x + 125y}{25} > \frac{2000}{25}\]This simplifies to:\[4x + 5y > 80\]

Step 4: Solve for \(y\) to Graph

To graph the inequality, solve for \(y\):\[5y > -4x + 80\]Divide every term by 5:\[y > -\frac{4}{5}x + 16\]

Step 5: Graph the Inequality

Draw the line \(y = -\frac{4}{5}x + 16\) on a coordinate plane. This line represents the border where \(4x + 5y = 80\). Use dotted lines because the inequality does not include \(\geq\). Shade the region above this line, as we need \(y\) to be greater for the inequality \(y > -\frac{4}{5}x + 16\).

Step 6: Identify Ordered-Pair Solutions

Find three suitable ordered-pair solutions \((x, y)\) that fall in the shaded region. Examples could include:1. \((x = 10, y = 12)\)2. \((x = 15, y = 8)\)3. \((x = 5, y = 14)\)Verify each solution by checking if \(4x + 5y > 80\).

Key Concepts

Linear Inequalities

Linear inequalities are expressions that involve a linear relationship using inequality symbols such as \( >, <, \geq, \leq \).
They are similar to linear equations but instead of an equal sign, they incorporate an inequality which shows a range of possible solutions.
For example, in the exercise, we have the inequality \(4x + 5y > 80\). This sets a relationship between the number of maple trees \(x\) and pine trees \(y\).
Unlike equations that provide a specific solution, inequalities describe a range of possible values that satisfy the condition.
When you solve a linear inequality, the aim is to find all values of the variables that make the inequality true:

• The inequality \(4x + 5y > 80\) tells us that combinations of \(x\) and \(y\) must result in a number greater than 80 when plugged into the expression.
• The inequality can be manipulated using similar rules to equations, but remember when you multiply or divide by a negative number, the inequality sign flips.

The solutions could be plotted on a graph for a more visual understanding, which leads us to the next topic.

Graphing Inequalities

Graphing inequalities involves plotting a line on a coordinate plane to show the solutions to the inequality.
It is a great way to visualize the relationship between variables.
The process starts by drawing the borderline, which you obtain from the inequality if it were an equation. For our inequality, the border is \(y = -\frac{4}{5}x + 16\):

• Instead of a solid line, use a dotted line for borderlines representing inequalities like \(>\) or \(<\) because the line itself is not part of the solution.
• Shade the area above the line to indicate where \(y\) is greater than \(-\frac{4}{5}x + 16\), as the inequality suggests.

By shading the appropriate region on the graph, we highlight all possible solutions to the linear inequality.
The points in this shaded area are the ordered-pair solutions like \((10, 12)\) or \((15, 8)\) that satisfy \(4x + 5y > 80\).

Problem-Solving in Mathematics

Problem-solving in mathematics often involves understanding the problem, then translating it into a mathematical language.
For inequalities, this means identifying what the problem is asking and converting it into an inequality statement.
Steps we follow generally include:

• Identifying the variables and constants in the problem scenario. In this exercise, it was the type of trees and their costs.
• Expressing the relationship using mathematical operations. Here, it is the cost inequalities.
• Simplifying or manipulating the inequality to make it easier to work with, if necessary.
• Graphing the inequality to visually determine solutions.
• Checking specific solutions within the context to ensure they satisfy the inequality statement.

These steps help in methodically approaching and solving mathematical problems effectively, ensuring all possible solutions are accurately found.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations.
They are the building blocks of mathematics used to create equations and inequalities.
In the scenario of the Arbor Day sale, expressions like \(100x\) and \(125y\) represent the revenue from trees.
Algebraic expressions convey relationships and can be modified or simplified as seen when we converted \(100x + 125y\) to \(4x + 5y\).
Some key points to remember about algebraic expressions in inequalities:

• They can be added, subtracted, multiplied, or divided but remember the special rule about inequalities: the direction changes when operated on by a negative number.
• Simplification can make working with expressions and inequalities more manageable, as seen when the expression was divided by 25.
• They model real-world situations, translating scenarios like sales and expenses into equations or inequalities that can be analyzed mathematically.

The ability to form and manipulate algebraic expressions is crucial for successful problem-solving in mathematics, especially in the context of linear inequalities.