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Chapter 1: Problem 2

Write the given statement as an inequality. $$ 4 y \text { is negative } $$

Short Answer

Expert verified
The inequality is \(4y < 0\).

Step by step solution

01

Identify the Mathematical Expression

The given statement is '4y is negative'. Here, the expression involves multiplying the variable \(y\) by \(4\).
02

Determine the Meaning of 'Negative'

In mathematics, 'negative' means less than zero. We want the expression \(4y\) to be less than zero.
03

Formulate the Inequality

To express the statement '4y is negative' as an inequality, write it as: \(4y < 0\).

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Key Concepts

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

Understanding Mathematical Expressions
Mathematical expressions are the building blocks of algebra. An expression is a combination of numbers, variables, and mathematical operations. In this problem, the expression is the product of the number 4 and the variable \( y \). Expressions don't include equality or inequality signs, but they often serve as the left or right side of an equation or inequality.
  • Expressions can involve various operations like addition, subtraction, multiplication, or division.
  • They can represent simple values or more complex computations.

In the given exercise, '4y' is a simple expression, indicating that you multiply the variable \( y \) by 4. It's crucial to clearly identify parts of an expression and understand what each component represents. This understanding becomes essential when transforming statements into mathematical equations or inequalities.
The Role of Variables in Inequalities
Variables are symbols used to represent unknown values. In mathematics, they are letters like \( x \), \( y \), or \( z \), and they help express general relationships. Understanding variables is crucial in working with inequalities, as they often determine the values that make an inequality true or false.
  • Variables allow for a wide range of solutions in inequalities.
  • They can represent real numbers, integers, or any other specified set of numbers.

In this situation, the inequality \( 4y < 0 \), the variable \( y \) represents values that when multiplied by 4, result in a negative number. Without a variable, mathematical models and equations would lack the ability to generalize and solve for unknowns, limiting problem-solving capabilities.
Dealing with Negative Numbers
Negative numbers are numbers less than zero. They are essential in mathematics, as they help in understanding concepts like debts, temperature below zero, or any situation involving a deficit. In inequalities, negative numbers can specify constraints or conditions that must be met.
  • Negative numbers are represented with a minus sign, like -1, -2, or -3.
  • In an inequality, determining if a number is negative can guide the direction of the inequality sign.

In our inequality \( 4y < 0 \), the phrase 'is negative' indicates that the product of \( 4 \) and \( y \) must fall into the category of negative numbers. By understanding negative values and how they influence mathematical expressions, you can better analyze and solve inequalities, ensuring your solutions fit within the intended range.

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Most popular questions from this chapter

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Use rationalization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(\frac{25-t}{5-\sqrt{t}}\) (b) \(\lim _{t \rightarrow 25} \frac{25-t}{5-\sqrt{t}}\)

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