Problem 139 Fill in the blank with \(<,=\... [FREE SOLUTION]

91影视

Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 1: Problem 139

Fill in the blank with $<,=$ or $\rangle$. $-|3|$ _____ $-|-(-5)|$

Short Answer

Expert verified

The answer is $>$.

Step by step solution

Evaluate Absolute Values

First, evaluate the absolute values in both terms of the inequality. Absolute value denotes the distance of a number from zero, regardless of direction, so $|3| = 3$ and $ |-(-5)| = 5$.

Apply Negative Signs

Next, apply the negative signs to these absolute values. The expression becomes: $-|3| = -3$ and $-|-(-5)| = -5$.

Compare the Two Calculated Values

Compare the two values from the last step: $-3$ and $-5$. Since $-3 > -5$, we fill the blank with $>$.

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Key Concepts

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

Absolute Values

When dealing with absolute values, it's essential to understand that absolute value measures how far a number is from zero, on a number line, without considering its direction. Thus, for any number $x$, the absolute value, expressed as $|x|$, is always non-negative. This means:

$|3| = 3$ because 3 is 3 units away from 0.
$|-5| = 5$ because -5 is 5 units away from 0. Notice the absolute value ignores the negative sign.

Absolute values are crucial in solving inequalities, as they give us a consistent base to compare numbers.

Negative Numbers

Negative numbers, those less than zero, are critical in understanding differences in magnitude. They can be thought of as a reflection on the opposite side of zero on the number line. In our problem:

The absolute value of 3, when negated, becomes $-|3| = -3$.
Similarly, the absolute value of -5, when negated, becomes $-|-(-5)| = -5$.

It's crucial to apply negative signs correctly after evaluating absolute values, as each negative sign changes the direction on the number line.

Comparative Expressions

Comparative expressions help us understand the relation between two numbers. In this context, we're looking at $-3$ and $-5$ to identify which is greater:

$-3$ is closer to zero on the number line than $-5$.
Therefore, $-3 > -5$.

Thus, the correct inequality sign to use is $>$. Understanding comparative expressions allows us to solve inequalities and compare magnitudes accurately.

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91影视

Fill in the blank with \(<,=\) or \(\rangle\). \(-|3|\) _____ \(-|-(-5)|\)

Short Answer

Step by step solution

Evaluate Absolute Values

Apply Negative Signs

Compare the Two Calculated Values

Key Concepts

Absolute Values

Negative Numbers

Comparative Expressions

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