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Chapter 5: Problem 6

Write an inequality to represent the given statement. The value of \(x\) is at least \(\frac{2}{3}\) the value of \(y\).

Short Answer

Expert verified
\( x \geq \frac{2}{3} y \)

Step by step solution

01

Understand the Inequality Statement

The statement 'The value of \(x\) is at least \(\frac{2}{3}\) the value of \(y\)' means that \(x\) is greater than or equal to \(\frac{2}{3}\) of \(y\).
02

Translate the Statement into an Inequality

We need to convert the verbal statement into a mathematical inequality. The phrase 'at least' indicates that \(x\) is greater than or equal to a certain value. This can be written as: \[ x \geq \frac{2}{3} y \]
03

Write the Final Inequality

Based on the translation in step 2, the inequality representing the given statement is: \[ x \geq \frac{2}{3} y \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

inequality translation
Translating verbal statements into algebraic inequalities is a vital skill in algebra. Understanding the key phrases and their mathematical equivalents helps you represent word problems correctly. For example, the phrase 'at least' signifies a lower boundary. Hence, 'The value of \(x\) is at least \(\frac{2}{3}\) the value of \(y\)' means \(x\) is greater than or equal to \(\frac{2}{3}y\). Always pinpoint these phrases, and identify the variables. Then, correctly translate them into an algebraic expression.

This step involves recognizing:
  • 'at least' as \(\geq\) symbol
  • identifying the variables and their specific relationship
  • writing the equivalent inequality for clear understanding
Breaking down the sentence correctly can help you translate any verbal inequality into a precise algebraic form.

algebraic expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. In this problem, \(\frac{2}{3}y\) is an algebraic expression, as it incorporates a variable \(y\) and a fraction. Understanding how to manipulate and use these expressions is essential for forming inequalities. It鈥檚 crucial to:
  • Identify the variables involved (like \(x\) and \(y\))
  • Understand the coefficients and constants (e.g., \(\frac{2}{3}\))
  • Combine them appropriately using mathematical operations

By mastering algebraic expressions, you'll be able to represent complex relationships and behaviors using symbols and numbers effectively.
inequality symbols
Inequality symbols are fundamental in expressing the relationship between two quantities. The most common symbols you鈥檒l encounter are:
  • \(>\) (greater than)
  • \(<\) (less than)
  • \(\geq\) (greater than or equal to)
  • \(\leq\) (less than or equal to)

In the exercise, the symbol \(\geq\) is used. It tells you that \(x\) is either greater than or exactly equal to \(\frac{2}{3}y\). This is essential in accurately interpreting and solving inequalities. Choose the correct symbol based on the context of the problem to represent the precise relationship between quantities.

Another key point is to check your translated inequality to ensure it logically matches the given statement. Freshly understanding these symbols can significantly alter the problem-solving accuracy in algebra.

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Most popular questions from this chapter

A college theater has a seating capacity of 2000 . It reserves \(x\) tickets for students and \(y\) tickets for general admission. For parts (a)-(d) write an inequality to represent the given statement. a. The total number of seats available is at most 2000 . b. The college wants to reserve at least 3 times as many student tickets as general admission tickets. c. The number of student tickets cannot be negative. d. The number of general admission tickets cannot be negative. e. Graph the solution set to the system of inequalities from parts (a)-(d).

Solve the system of equations by using the addition method. (See Examples \(3-4)\) $$ \begin{array}{l} 3 x-7 y=1 \\ 6 x+5 y=-17 \end{array} $$

Solve the system using any method. $$ \begin{array}{l} 2 x=\frac{y}{2}+1 \\ 0.04 x-0.01 y=0.02 \end{array} $$

Solve the system of equations by using the addition method. (See Examples \(3-4)\) $$ \begin{array}{l} -3(x-y)=y-14 \\ 2 x+2=7 y \end{array} $$

A plant nursery sells two sizes of oak trees to landscapers. Large trees cost the nursery \(\$ 120\) from the grower. Small trees cost the nursery \(\$ 80\). The profit for each large tree sold is \(\$ 35\) and the profit for each small tree sold is \(\$ 30 .\) The monthly demand is at most 400 oak trees. Furthermore, the nursery does not want to allocate more than \(\$ 43,200\) each month on inventory for oak trees. a. Determine the number of large oak trees and the number of small oak trees that the nursery should have in its inventory each month to maximize profit. (Assume that all trees in inventory are sold.) b. What is the maximum profit? c. If the profit on large trees were \(\$ 50\), and the profit on small trees remained the same, then how many of each should the nursery have to maximize profit?
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