91Ó°ÊÓ

Understanding absolute value is critical when dealing with negative numbers, especially when you're required to order or compare them. The absolute value of a number is the distance of that number from zero on the number line, without considering the direction. This means it's always a non-negative value.

For instance, the absolute value of -7, denoted as \( |-7| \), is 7. This is because whether you move 7 units to the left or to the right of zero on the number line, the distance you've moved is still 7 units. Similarly, the absolute value of a positive number, like 4 (represented as \( |4| \)), is also 4.

• The absolute value turns negative numbers into their positive counterparts.
• It helps in determining magnitude without direction.
• Absolute values are used to simplify expressions before further operations such as ordering.

Understanding absolute value is essential for correctly interpreting negative expressions and ordering numbers accurately.

Negative integers are the set of integers less than zero. They are represented with a minus sign (-) and appear to the left of zero on a horizontal number line.

Working with negative integers can be tricky, but keeping a few points in mind can help:

• When comparing negative numbers, the number closer to zero is considered larger. For example, -3 is larger than -5.
• The more negative the number, the smaller it is considered. So, -9 is smaller than -7.
• Subtracting a negative number is the same as adding its positive counterpart (subtracting -3 is the same as adding 3).

When it comes to our exercise, evaluating -|-7| results in -7 because we're essentially finding the absolute value (7) and then applying the negative sign, making it a negative integer.

Ordering numbers in ascending order means arranging them from the smallest value to the largest. When the numbers include negatives, it's crucial to recognize that negative integers are smaller than positive integers and that among negative integers, those with a larger absolute value are actually smaller.

To arrange numbers in ascending order, identify the smallest number first, which is the number farthest to the left on the number line if they were plotted. For positive numbers, this technique is straightforward. But with the inclusion of negatives, always remember that a number with a larger negative value (like -9) is actually smaller than a number with a smaller negative value (like -7).

In the context of the exercise we're looking at, -9 being the smallest negative integer (farthest from zero on the left side of the number line) is placed first, followed by -7, then -5, and finally the positive integer 4.