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Chapter 0: Problem 13

\(9-16\) State whether each inequality is true or false. $$ -\pi>-3 $$

Short Answer

Expert verified
False; the inequality is incorrect.

Step by step solution

01

Understanding the Symbols

First, we need to understand what the inequality and mathematical symbols mean. The inequality symbol '>' signifies 'greater than.' \( \pi \) (pi) is a constant approximately equal to 3.14159, and '-\pi' means we are considering the negative of this constant.
02

Substitute the Approximate Value

Next, substitute the approximate value of \( \pi \) to understand the inequality. \( \pi \approx 3.14159 \), so \( -\pi \approx -3.14159 \).
03

Compare the Values

Now, compare the two values: \(-3.14159\) and \(-3\). Since \(-3.14159\) is less than \(-3\), the inequality \(-\pi > -3\) is incorrect, meaning it is false.

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Key Concepts

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

Mathematical Symbols
In mathematics, symbols are used to represent ideas or operations in a concise way, making them universal and efficient. Some common symbols include numbers, operators like the plus sign '+', and inequality symbols such as '>' and '<'.

An inequality symbol is crucial in showing the relationship between two values. The symbol '>' means 'greater than', indicating that the number on the left is larger than the number on the right. Conversely, '<' denotes 'less than'.
  • Understanding how these symbols work is essential for solving inequality problems.
  • They form the foundation for equations and comparisons in mathematics.
Inequalities also use other symbols like '鈮�' for 'greater than or equal to', and '鈮�' for 'less than or equal to'.
Constant Values
Constant values are fixed numbers that do not change. Some well-known constants in mathematics include \( \pi \), which is approximately equal to 3.14159. Constants are important because they provide exact values that are used repeatedly in calculations and equations.

When dealing with inequalities, constants are used to give a specific reference point. For example, in the inequality \( -\pi > -3 \), \( \pi \) is the constant. By knowing approximations like \( \pi \approx 3.14159 \), we can determine the approximate value of expressions like \( -\pi \).
  • This helps in substituting into inequalities to make comparisons and conclusions.
  • Constants provide precision and context to mathematical problems.
Comparison of Values
The process of comparing values involves evaluating their size relative to one another. This is fundamental in determining the truth value of inequalities.

To compare values in inequalities, substituting numerical approximations is often necessary. For instance, calculating -蟺 using the approximate value of 蟺 (which is 3.14159) gives -3.14159. Now, you need to determine if -3.14159 is greater than -3.

Comparing these negative numbers, we see that -3.14159 is actually less than -3, because on the number line, negative numbers that are further from zero are smaller.
  • This results in the original statement being false.
  • Understanding the relative position of numbers, especially negatives, is important in these comparisons.
Thus, the comparison of values helps verify and decide the accuracy of mathematical statements.

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Most popular questions from this chapter

Perform the multiplication or division and simplify. $$ \frac{t-3}{t^{2}+9} \cdot \frac{t+3}{t^{2}-9} $$

Perform the addition or subtraction and simplify. $$ \frac{1}{x+1}-\frac{1}{x+2} $$

Rationalize the denominator. $$ \frac{y}{\sqrt{3}+\sqrt{y}} $$

Simplify the expression. (This type of expression arises in calculus when using the 鈥渜uotient rule.鈥�) $$ \frac{\left(1-x^{2}\right)^{1 / 2}+x^{2}\left(1-x^{2}\right)^{-1 / 2}}{1-x^{2}} $$

Perform the addition or subtraction and simplify. $$ \frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}} $$
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