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Chapter 1: Problem 82

Determine whether each statement is true or false. \(-|12|>|-15|\)

Short Answer

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True

Step by step solution

01

Evaluate the Absolute Value

Calculate the absolute value of 12. Since \(|12| = 12\), the expression becomes \(-12\).
02

Simplify the Right-Hand Side

The right-hand side of the inequality is \(-15\).
03

Compare the Values

Now compare \(-12\) and \(-15\). Since \(-12\) is greater than \(-15\), the statement \(-|12| > -15\) is true.

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Key Concepts

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

absolute value
The absolute value of a number is its distance from zero on a number line, regardless of direction. To find the absolute value, you simply disregard the sign.
For example:
  • The absolute value of 12 is \(|12| = 12\) since it's 12 units away from zero.
  • Likewise, the absolute value of -12 is \(|-12| = 12\).
Absolute value turns all numbers into non-negative values. This concept is essential in mathematics, especially when dealing with distances and magnitudes.
inequality
Inequality compares two values, showing if one is less than, greater than, or equal to another. In mathematical notation:
  • \(a < b\) means 'a' is less than 'b'.
  • \(a > b\) means 'a' is greater than 'b'.
  • \(a \leq b\) means 'a' is less than or equal to 'b'.
  • \(a \geq b\) means 'a' is greater than or equal to 'b'.
In our example, we used the inequality \(-|12| > -15\). We simplified the left side to \(-12\) and then compared \(-12\) with \(-15\). Since \(-12\) is indeed greater than \(-15\), the inequality holds true.
negative numbers
Negative numbers are numbers less than zero, represented with a 鈥�-鈥� sign. Understanding them is crucial for comparisons especially in inequalities.
  • On a number line, negative numbers lie to the left of zero.
  • For example, -12 is 12 units to the left of zero.
When comparing negative numbers:
  • A larger negative number (closer to zero) is always greater than a smaller one (further from zero). Hence, \(-12 > -15\).
This concept helps us understand why in our exercise, \(-12 > -15\) and concludes our statement as true.

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Most popular questions from this chapter

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Solve each problem. In 2005, Americans saved \(-0.5 \%\) of their after-tax incomes. In September 2017 , they saved \(3.1 \%\). (Data from U.S. Bureau of Economic Analysis.) (a) Express the difference between these amounts as a positive number. (b) How could Americans have a negative personal savings rate in \(2005 ?\)

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