Problem 46 Simplify each of the following e... [FREE SOLUTION]

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Prealgebra

David Arnold

$Math Studyset 91影视 Explanations$ Math

2020 Edition

Chapter 2: Problem 46

Simplify each of the following expressions. . $|-1|$.

Short Answer

Expert verified

The simplified expression is 1.

Step by step solution

Understand the Absolute Value Function

The expression $|-1|$ involves the absolute value function. The absolute value of a number is its distance from zero on the number line, without considering direction. For any real number $x$, the absolute value is defined as follows: $|x| = x$ if $x \geq 0$ and $|x| = -x$ if $x < 0$.

Apply the Definition to the Expression

Since $x = -1$ in the expression $|-1|$, we need to check the condition $x < 0$. Given that $-1$ is less than zero, use the rule $|x| = -x$. Therefore, $|-1| = -(-1)$.

Simplify the Expression

Calculate the expression $-(-1)$, which translates to the opposite of $-1$. The opposite of $-1$ is $+1$ or simply \1\.

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Key Concepts

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

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are usually represented with a minus sign (-) in front of them, such as -1, -2, and -3.
These numbers are placed on the left side of zero on the number line, because they are smaller than zero.
Instead of gaining value as they go further from zero, negative numbers represent a decrease in value.

Example: -5 is five units away from zero and is less than -3, since -5 is positioned further left on the number line.
Real-world scenarios include elevations below sea level or temperatures below freezing.

Working with negative numbers requires understanding their position in comparison to zero in mathematical expressions.

Using the Number Line

A number line is a visual tool that helps us understand how numbers are organized. On a typical number line, zero is positioned at the center.
Positive numbers spread out to the right of zero while negative numbers are placed to the left.
When interpreting absolute values, the number line is extremely helpful. Absolute value measures the distance of any number from zero.

For instance, on the number line, both 1 and -1 are one unit away from zero, making their absolute values equal (|1| = |-1| = 1).
This allows us to ignore the negative sign as it only indicates direction, not distance.

By understanding a number line, students can better visualize math operations involving both positive and negative numbers.

Simplifying Expressions

Simplifying expressions involves breaking down expressions into their most basic form. In the case of absolute values, simplification often means finding the simplest, and most straightforward representation.
The absolute value $|x|$ is a perfect example. When simplifying $|-1|$, we aren't actually looking for a new value, but instead, its simplest form ()1\). This step-by-step simplification requires applying rules:

If the number inside the absolute value is negative, like -1, you remove the negative by multiplying by \(-1)\ to yield a positive result.
This is confirmed mathematically as: $-(-1) = +1$.

Understanding how to simplify expressions by following set rules adds clarity and accuracy to problem-solving processes.

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91影视

Simplify each of the following expressions. . \(|-1|\).

Short Answer

Step by step solution

Understand the Absolute Value Function

Apply the Definition to the Expression

Simplify the Expression

Key Concepts

Understanding Negative Numbers

Using the Number Line

Simplifying Expressions

One App. One Place for Learning.

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