Problem 110 - Prealgebra and Introductory Algebra

Question

Write the given numbers in order from smallest to largest. $$|-8|,-(-3),|2|,-|-5|$$

Step-by-Step Solution

Answer

-5, 2, 3, 8

Step 1: Compute Absolute Values and Negative Expressions

Firstly, compute the absolute values and negative expressions for each number. The absolute value of -8 is 8, -(-3) is 3, the absolute value of 2 is 2, and -|-5| is -5.

Step 2: Arrange the Numbers from Smallest to Largest

After computing, we have the set of numbers {8, 3, 2, -5}. These numbers should be put in order from smallest to largest. Thus, we get -5, 2, 3, 8.

Key Concepts

Absolute value

When we talk about the absolute value of a number, we're interested in how far that number is from zero on the number line.
Absolute value is always a positive number or zero. To determine it, simply ignore any negative sign in front of a number and treat it as positive.

For example:

• The absolute value of \(-8\) is \(|-8| = 8\).
• Similarly, the absolute value of 5 is \(|5| = 5\).
• And \(|0| = 0\) since zero is zero distance from itself on the number line.

Absolute value can change how we view numbers in terms of distance from zero, and it has a key role in ordering numbers because it converts everything to non-negative values for comparisons.

Negative numbers

Negative numbers are those less than zero and are located to the left of zero on the number line.
They are usually represented with a minus sign. Working with negative numbers involves understanding some special rules.

Consider these points:

• Multiplying or dividing two negative numbers results in a positive number (e.g., \((-2) \times (-3) = 6\)).
• Adding two negative numbers keeps the result negative and shifts further left on the number line (e.g., \(-2 + (-3) = -5\)).
• Subtracting a negative number is like adding its positive counterpart (e.g., \(-(-3) = 3\)).

Understanding these simple operations makes it easier to solve and arrange problems with negative numbers.

Ordering numbers

Ordering numbers involves arranging them from smallest to largest or vice versa, and it's a fundamental skill in math.
When you have a mix of positive, negative, and absolute values, the task can seem tricky at first glance.

Here's a simplified approach:

• Convert all absolute values to reveal their true magnitude without a sign.
• Identify any negative expressions and solve them to understand their actual value.
• Once all numbers are in their proper numerical form, compare them and put them in order.

For example, given numbers \(|-8|, -(-3), |2|, -|-5|\), compute as follows:

• \(|-8| = 8\)
• \(-(-3) = 3\)
• \(|2| = 2\)
• \(-|-5| = -5\)

Then, order as: -5, 2, 3, 8. This process can be applied to any set of numbers.