Question

Place the correct symbol \((<,>\), or \(=)\) between the two real numbers.\(\begin{array}{ll}-5 & -|5|\end{array}\)

Short answer

-5 = -|5|

Step-by-step solution

Solve the absolute value

First, calculate the absolute value of 5, as absolute value is the distance a number is from zero. We denote this |5| which is equal to 5.

Subtract the absolute value

Next, it's time to subtract the absolute value from zero, which gives -|5|. From step 1 we know |5| equals to 5, therefore -|5| equals to -5.

Compare the two numbers

Finally, the task is to compare -5 and -|5| which equals to -5. Therefore, -5 = -5, because they are exactly the same.

Key concepts

Absolute Value

Understanding the concept of absolute value is crucial to mastering expressions involving real numbers. The absolute value of a number is fundamentally the distance of that number from zero on a number line, without considering direction. Regardless of whether a number is positive or negative, its absolute value is always a non-negative number. For instance, the absolute value of both 5 and -5 is 5, which we denote as \(|5| = 5\) and \(|-5| = 5\).

Imagining the number line, absolute value signifies how far a number is from the origin (zero), not the direction in which it lies. When solving problems that include absolute value, it's important to remember this intrinsic definition, as it helps simplify and correctly interpret expressions and inequalities that involve absolute values.

Inequalities

When dealing with inequalities, we are looking at the relative size of two numbers and determining which one is greater, lesser, or if they are equal. Inequalities are denoted by signs such as \(<\) for 'less than', \(>\) for 'greater than', and \(=\) for 'equal to'. These symbols serve as a shorthand for comparing numeric values or expressions.

It is particularly important to pay attention to the direction of these symbols, as they indicate the flow of the numerical comparison. For instance, if we have \(7 > 5\), it means that 7 is greater than 5. Conversely, \(3 < 5\) indicates that 3 is less than 5. Dealing with inequalities requires an understanding of the number line's order and how to manipulate expressions while preserving the inequality's correctness, especially when negative numbers and absolute values are involved.

Number Comparison

Comparing numbers is a fundamental skill in mathematics, encompassing the recognition of numerically larger, smaller, or equivalent values between two numbers. The comparison often relies on inherent understanding of the number line and the position of numbers upon it. In our exercise, we are asked to compare two real numbers using our knowledge of absolute values and properties of equality and inequality.

When comparing numbers, it's also crucial to consider special mathematical properties and symbols used. For instance, the expression \(-5 = -|5|\) in our example, shows that when the absolute value of a number is used within a negative expression, it can be equivalent to its negative counterpart without the absolute value - both represent the same point on the number line. This concept allows for simplification and accurate comparison of complex numerical expressions.