Question

Place the correct symbol $(<,>$, or $=)$ between the two real numbers.$|-7||7|$

Short answer

|-7| = |7|

Step-by-step solution

Evaluate |-7|

The absolute value, noted by the modulus sign '|', of -7 is 7. This is because the absolute value of a number is its distance from zero and is always a positive number or zero.

Evaluate |7|

The absolute value of 7 is also 7. This is because 7 is already a nonnegative number, and the absolute value of a nonnegative number is the number itself.

Compare the two values

After evaluating, both |-7| and |7| are equal to 7. Comparing them, we see that they are equal.

Key concepts

Understanding Real Numbers

Real numbers are the set of all rational and irrational numbers. They include numbers like integers, fractions, and decimals. Essentially, any number you can find on the number line is a real number. Real numbers can be either positive, negative, or zero.
Real numbers are useful in many everyday calculations, from simple arithmetic to more complex mathematics, like calculus. When dealing with real numbers, especially in equations, it’s crucial to comprehend how each number behaves. For example, negative numbers have opposites that lie the same distance from zero but in the opposite direction.

Exploring Modulus or Absolute Value

The modulus or absolute value of a number tells us how far away the number is from zero without considering its direction. It is always nonnegative.
When you see a number inside vertical bars like $|-7|$, it means you take the absolute value. So, $|-7|$ becomes 7 because the distance from zero is 7, ignoring the negative sign. Likewise, $|7|$ is also 7 because it's already positive.

• Absolute value gives us a clearer idea of size without the direction.
• It helps in understanding the magnitude of numbers irrespective of their sign.

Comparison of Numbers Using Absolute Value

Comparing numbers is essential in understanding how they relate on the number line. When using absolute values for comparison, you focus on their magnitude rather than sign or direction.
In the exercise, $|-7|$ and $|7|$ both evaluate to 7. By comparing their absolute values, you see they are equal.

• Use <, >, or = to show the relationship between numbers.
• Equality of absolute values indicates the same distance from zero.

This method is helpful when comparing negative numbers to each other or to positive ones, ensuring you understand size over sign.