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Chapter 1: Problem 72

Decide whether each statement is true or false. $$ -|12|>|-15| $$

Short Answer

Expert verified
True

Step by step solution

01

Evaluate Absolute Values

Determine the absolute value of each number. The absolute value of 12 is 12, so \( |12| = 12 \). The absolute value of -15 is 15, so \( |-15| = 15 \).
02

Apply Negative Sign

Apply the negative sign to each absolute value. This results in \( -|12| = -12 \) and \( -|15| = -15 \).
03

Compare the Values

Compare the resulting values: -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 concept of absolute value is fundamental in mathematics. Absolute value refers to the distance of a number from zero on the number line, regardless of direction. This means that both positive and negative numbers have positive absolute values. For instance, the absolute value of 12 is written as \( |12| \), and it equals 12. Similarly, the absolute value of -15 is written as \( |-15| \), and it equals 15. This concept helps in simplifying and solving equations involving both positive and negative numbers. Absolute values are always non-negative.
negative numbers
Negative numbers represent values less than zero and are denoted with a minus sign (e.g., -15). They are different from positive numbers and zero, and they play a crucial role in mathematics. When dealing with negative numbers in expressions, especially those involving absolute values, it is important to apply the negative sign after determining the absolute value. For example, evaluating \( -|12|\) means taking the absolute value of 12 (which is 12) and then applying the negative sign, resulting in -12. This ensures clarity and accuracy in mathematical operations.
comparisons
Comparing numbers is a key skill in mathematics. When comparing negative numbers, remember that the number closer to zero is considered larger. For example, -12 is greater than -15 because -12 is closer to zero. This can be counterintuitive since we often think of larger numbers as greater, but with negatives, it's reversed. Therefore, in our example, since -12 is greater than -15, the statement \( -|12| > |-15| \) is true.
step-by-step solution
Solving mathematical problems step-by-step ensures accuracy and helps you understand each part of the process. Here鈥檚 a detailed look at the solution to the given problem:
  • Step 1: Determine the absolute values. For the numbers given, \( |12| = 12 \) and \( |-15| = 15 \).
  • Step 2: Apply the negative sign. This gives us \( -|12| = -12 \) and \( -|15| = -15 \).
  • Step 3: Compare the results. Compare -12 and -15. Since -12 is greater than -15, the statement \( -|12| > -|15| \) is true.
This approach breaks down the problem, ensuring each step is clear and logical. By working through problems methodically, you'll build a stronger understanding of mathematical concepts.

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