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Chapter 1: Problem 91

Write each word statement in symbols. One-half is less than or equal to two-fourths.

Short Answer

Expert verified
\frac{1}{2} \leq \frac{2}{4}

Step by step solution

01

- Identify the Numerical Values

Identify the numerical values given in the word statement: one-half and two-fourths.
02

- Write the Numerical Values as Fractions

Convert the word statements 'one-half' and 'two-fourths' into fractional notation: \(\frac{1}{2}\) and \(\frac{2}{4}\) respectively.
03

- Understand the Inequality Statement

Identify the inequality relationship being described: 'is less than or equal to.' This corresponds to the symbol \(\leq\).
04

- Combine the Fractions and Inequality Symbol

Combine the fractions and the inequality symbol into one mathematical expression to represent the word statement: \(\frac{1}{2} \leq \frac{2}{4}\).

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Key Concepts

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

Inequalities
Inequalities are mathematical expressions that show the relationship between two values, where one value is either less than, greater than, less than or equal to, or greater than or equal to the other.

In this exercise, we use the inequality symbol \( \leq \), which means 'less than or equal to.'
This symbol tells us that the value on the left side is either smaller than or the same as the value on the right side.

In practical terms:
  • If you have \( x \leq y \), then \( x \) can be either smaller than \( y \) or exactly equal to \( y \).


Understanding inequalities is crucial because they can help to describe a range of possible values rather than just a single value. Knowing how to read and use these expressions allows us to tackle a broader set of mathematical problems.
Fractions
Fractions represent parts of a whole. In this exercise, we deal with the fractions 'one-half' and 'two-fourths.'

The fraction for one-half is written mathematically as \( \frac{1}{2}\). This means one part out of two equal parts.

Similarly, the fraction for two-fourths is written as \( \frac{2}{4} \), which means two parts out of four equal parts.
  • To simplify fractions, divide both the numerator and the denominator by their greatest common divisor. In this example, \ \frac{2}{4}\ becomes \ \frac{1}{2}\ because the greatest common divisor of 2 and 4 is 2.


Understanding fractions is essential as they are used in many areas, including algebra, geometry, and real-life situations like cooking and finances.
Mathematical Notation
Mathematical notation is a system of symbols used to write mathematical statements clearly and concisely.

In this exercise, converting a word statement into mathematical symbols makes the problem easier to analyze and solve.

For instance:
  • The word statement 'one-half is less than or equal to two-fourths' can be translated into \( \frac{1}{2} \leq \frac{2}{4} \).


This notation communicates the same information more precisely.

Fractions and inequalities are examples of fundamental mathematical notations.
  • Fractions, like \( \frac{1}{2}\), represent parts of a whole.

  • Inequality symbols, like \( \leq\), show relationships between quantities.


Mastering these notations helps to build a solid foundation in mathematics, making it easier to understand complex topics later on.

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Most popular questions from this chapter

Write a numerical expression for each phrase and simplify. The sum of \(-4\) and \(-10,\) increased by 12

Decide whether each statement is true or false. \(-|-12| \leq-|-15|\)

Solve each problem. In \(1998,\) undergraduate college students had an average (mean) credit card balance of 1879. The average balance increased \(\$ 869\) by 2000 , then dropped \(\$ 579\) by \(2004,\) and then increased \(\$ 1004\) by \(2008 .\) What was the average credit card balance of undergraduate college students in \(2008 ?\)

Find each difference. $$ \frac{1}{2}-\left(-\frac{1}{4}\right) $$

In August, Susan Goodman began with a checking account balance of \(\$ 904.89\). Her checks and deposits for August are as follows: $$ \begin{array}{|r|r|} \hline \text { Checks } & \text { Deposits } \\ \hline \$ 35.84 & \$ 85.00 \\ \$ 26.14 & \$ 120.76 \\ \$ 3.12 & \\ \hline \end{array} $$ Assuming no other transactions, what was her account balance at the end of August?
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