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Chapter 1: Problem 74

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

Short Answer

Expert verified
False

Step by step solution

01

- Simplify the Absolute Values

First, compute the absolute values for both numbers inside the inequality. Recall that the absolute value of a number is its distance from zero, regardless of direction. Therefore, \( |-12| = 12 \) and \( |-15| = 15 \).
02

- Apply Negative Signs

Next, apply the negative signs outside the absolute values. So, \( -|-12| = -12 \) and \( -|-15| = -15 \).
03

- Compare the Results

Compare the two results from the previous step. The inequality given is \( -12 \leq -15 \). Observe that \( -12 \) is not less than or equal to \( -15 \). In fact, \( -12 \) is greater than \( -15 \).
04

- Conclude the Inequality

Since \( -12 \) is greater than \( -15 \), the initial statement \( -|-12| \leq -|-15| \) is false.

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Key Concepts

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

absolute value
Absolute value is a fundamental concept in algebra and mathematics. It represents the distance of a number from zero on a number line, ignoring any negative sign. For example, the absolute value of -12 is written as \(|-12|\) and is equal to 12.
Similarly, the absolute value of -15 is \(|-15|\) and equals 15.
Think of absolute value as measuring pure distance:
  • It only cares about how far a number is from zero.
  • It doesn't care if the number is positive or negative.

This is why \( |x| = x \) if x is positive, and \( |x| = -x \) if x is negative.
Understanding absolute value is crucial for solving many algebra problems, especially those involving inequalities.
inequality comparison
An inequality comparison involves checking if one number is greater than, less than, or equal to another number. In the exercise, we compared \(-|-12| \) and \(-|-15| \).
First, we simplified the absolute values:
  • \(|-12| = 12\)
  • \(|-15| = 15\)

Next, we applied the negative signs:
  • \(-|-12| = -12\)
  • \(-|-15| = -15\)

The final inequality was \(-12 \leq -15\).
To compare the results, think in terms of a number line:
  • -12 is closer to zero than -15
  • -12 is greater than -15
This means \(-12 \leq -15\) is false, since -12 is actually greater than -15.
Remember: on a number line, a larger negative number is always less than a smaller negative number.
negative numbers
Negative numbers are numbers less than zero. They are written with a minus sign (-). For example, -12 and -15 are negative numbers.
Comparing negative numbers can be a bit tricky at first:
  • The more negative a number, the smaller it actually is.
  • -15 is less than -12 because it is further from zero.

When applying inequalities such as \(\textless \), \(\textgreater \), or \(\textless \ =\), remember these tips:
  • -12 is greater than -15.
  • -20 is less than -5.

Negative numbers are crucial in mathematics, especially in regards to temperature, finance, and physics.
Understanding how they behave within inequalities and absolute values is essential for solving various math problems effectively.

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Most popular questions from this chapter

Find each difference. $$ \frac{9}{10}-\left(\frac{1}{8}-\frac{3}{10}\right) $$

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