Problem 6 - Algebra 1

Question

Complete the statement using \(>\) or \(<\). $$-2 \underline{?}-\frac{1}{2}$$

Step-by-Step Solution

Answer

-2 < -1/2

Step 1: Understand the Number Line

On a number line, numbers to the right are greater than those on the left. Positive numbers are to the right of zero, and the negative numbers are on the left. The larger the absolute value of a negative number, the smaller its actual value.

Step 2: Compare -2 and -0.5

Since -2 is to the left of -0.5 on the number line, we can say that -2 is less than -0.5.

Step 3: Fill in the symbol

The symbol for 'less than' is \(<\), so the correct completion of the statement should be '-2 < -1/2'.

Key Concepts

Number Line

A number line is a visual representation of numbers laid out in order, typically with zero in the center. It's like a ruler that helps us easily compare the size of any two numbers, whether they are positive or negative.

• Negative numbers are positioned on the left-hand side of the zero.
• Positive numbers are on the right-hand side.

The further you move to the right on the number line, the larger the values become. Conversely, moving to the left will lead you to smaller numbers.

To compare numbers on this line, simply see which is further to the right.

• A number to the left is always smaller than a number to the right.

Understanding this straightforward layout is key when working with inequalities like in our exercise.

Absolute Value

The concept of absolute value is crucial when dealing with negatives. Absolute value focuses only on how far a number is from zero, not in which direction on the number line.

• It's always a non-negative number.
• Represented as \(|x|\), where x is the original number.

For example, the absolute value of -3 is 3, because -3 is three spaces away from zero.

Though negative numbers and their absolute values can look different, absolute values help us compare how ‘big’ their magnitude is. Therefore, even though \(-2\) has an absolute value of 2 and -0.5 has an absolute value of 0.5, on a number line, -0.5 will actually be closer to zero, making \(-0.5\) greater than \(-2\).

This aspect is a great tool in solving inequality problems and sets the groundwork for many mathematical concepts.

Negative Numbers

Understanding negative numbers involves grasping their position on the number line and their relationship to positive numbers.

• Negative numbers are found to the left of zero.
• The more negative a number, the further left it is placed.

A greater negative number actually signifies a smaller value because you're moving farther from zero.

This is why -2 is less than -0.5 even though 2 is numerically larger than 0.5.

When comparing negative numbers, think of it as who is closer to the positive side of the number line. Here, -0.5 is closer to zero than -2, making it greater than -2.

• For quick decisions, remember a bigger absolute value in negatives means a smaller actual value.

Understanding this helps in performing operations like adding, subtracting, or even solving inequality problems in mathematics with ease.