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Chapter 5: Problem 5

Write an inequality to represent the given statement. The value of \(y\) does not exceed three times the value of \(x\).

Short Answer

Expert verified
The inequality is \( y \leq 3x \).

Step by step solution

01

Understand the Problem

Identify the relationship between the variables given in the statement. Here, it specifies that the value of y does not exceed three times the value of x.
02

Translate the Statement into a Mathematical Expression

The phrase 'does not exceed' can be represented by the inequality symbol \( \leq \). So, y does not exceed three times x can be written as: \( y \leq 3x \)
03

Write the Final Inequality

Combine the variables and the inequality symbol to form the final inequality: \( y \leq 3x \).

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Key Concepts

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

Inequality Symbols
Inequalities are expressions that show a relationship between two values. Instead of indicating that two values are equal, they show if one value is greater than, less than, or not equal to another. The main inequality symbols are:
  • < (\text{less than})
  • > (\text{greater than})
  • \text{<=} (\text{less than or equal to})
  • \text{>=} (\text{greater than or equal to})
  • eq (\text{not equal to})
In the given exercise, 'does not exceed' means 'is less than or equal to', which we represent with the symbol \text{\(\text{<=}\)}. This helps us translate words into mathematical language, making it easier to work with and solve.
Variables in Algebra
Variables are symbols used to represent unknown numbers in algebra. In our exercise, we see the variables x and y. These variables allow us to write equations and inequalities in a generalized form. Here鈥檚 how they work:
  • x typically stands for an independent variable (the one you can choose)
  • y stands for a dependent variable (its value depends on the value of x)
By expressing y in terms of x, we can explore their relationship. In this case, the inequality tells us how the value of y relates to three times the value of x.
Mathematical Representation
Mathematical representation is crucial for understanding and solving problems. It involves translating words into mathematical symbols and expressions. In the given exercise, the statement 'The value of y does not exceed three times the value of x' is translated into the inequality \text{\(y \text{<=} 3x\)}. This representation allows us to clearly see the relationship and apply algebraic methods to analyze it. Remember:
  • Identify keywords (like 'does not exceed')
  • Choose the appropriate inequality symbol (<= in this case)
  • Combine variables and symbols to form the inequality
This systematic approach helps in forming correct and useful mathematical expressions from word problems.

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Most popular questions from this chapter

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?

Find two numbers whose sum is 12 and whose product is 35 .

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When setting up a partial fraction decomposition, if the denominator of a fraction is a quadratic polynomial irreducible over the integers, then the numerator should be (constant/linear). That is, should the numerator be set up as \(A\) or \(A x+B ?\)

A sawmill cuts boards for a lumber supplier. When saws \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) all work for \(6 \mathrm{hr}\), they cut 7200 linear board-ft of lumber. It would take saws \(A\) and \(B\) working together \(9.6 \mathrm{hr}\) to cut \(7200 \mathrm{ft}\) of lumber. Saws \(\mathrm{B}\) and \(\mathrm{C}\) can cut 7200 ft of lumber in \(9 \mathrm{hr}\). Find the rate (in \(\mathrm{ft} / \mathrm{hr}\) ) that each saw can cut lumber.
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