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Chapter 5: Problem 1

Write an inequality to represent the given statement. The sum of \(x\) and \(y\) is at most 70 .

Short Answer

Expert verified
The inequality is \(x + y \leq 70\).

Step by step solution

01

Understand the problem

We need to write an inequality for the statement 'The sum of \(x\) and \(y\) is at most 70'. 'At most' means that the sum can be less than or equal to 70.
02

Represent the sum of x and y

Write the sum as an expression: \(x + y\).
03

Formulate the inequality

Since the sum is at most 70, we write the inequality as: \(x + y \leq 70\).

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Key Concepts

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

inequality representation
Inequalities are useful tools in algebra to compare expressions. They help us understand relationships between mathematical expressions. For example, 'at most' or 'no more than' suggests a limit is not exceeded.

In this exercise, the phrase 'at most' means that the sum should be either equal to or less than a specific value. This translates to an inequality using the symbol \( \leq \).

Remember, inequalities indicate the following types of relationships:
  • \(>\) : greater than
  • \(<\) : less than
  • \(\geq\) : greater than or equal to
  • \(\leq\) : less than or equal to
In our case, 'at most 70' is represented by \( \leq 70 \).
sum of variables
In algebra, the sum of variables often needs to be expressed concisely. When we talk about the 'sum of x and y,' we are referring to the addition operation.

The sum can be written as \( x + y \). Understanding how to write and interpret this form is crucial in forming equations and inequalities.

It's essential to follow the correct order of operations and use parentheses if needed to group terms correctly. For this example, the sum does not require any special grouping, making the expression straightforward.
mathematical expressions
Mathematical expressions are combinations of numbers, variables, and operations (like addition and multiplication) that represent values.

In creating or solving expressions, clarity and correctness are key. For instance, the expression \( x + y \) correctly combines the variables for their sum.

**Formulating Inequalities With Expressions**
To translate a verbal statement to a mathematical inequality, we must:
  • Identify key terms (e.g., 'at most', 'less than', 'greater than').
  • Understand the relationships implied (e.g., 'at most 70' becomes \( \leq 70 \)).
  • Combine variables and constants appropriately (e.g., \( x + y \)).
By following these steps, we create meaningful and accurate representations of mathematical ideas.

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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?

Solve the system. $$ \begin{array}{r} \log x+2 \log y=5 \\ 2 \log x-\log y=0 \end{array} $$

Two particles begin at the same point and move at different speeds along a circular path of circumference \(280 \mathrm{ft}\). Moving in opposite directions, they pass in \(10 \mathrm{sec} .\) Moving in the same direction, they pass in \(70 \mathrm{sec} .\) Find the speed of each particle.

A coordinate system is placed at the center of a town with the positive \(x\) -axis pointing east, and the positive \(y\) -axis pointing north. A cell tower is located \(4 \mathrm{mi}\) west and \(5 \mathrm{mi}\) north of the origin. a. If the tower has a 8 -mi range, write an inequality that represents the points on the map serviced by this tower. b. Can a resident 5 mi east of the center of town get a signal from this tower?

Solve the system using any method. $$ \begin{array}{l} 4(x-2)=6 y+3 \\ \frac{1}{4} x-\frac{3}{8} y=-\frac{1}{2} \end{array} $$
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