Problem 80 - Contemporary Precalculus

Question

Fill the blank with \(<,=\), or \(>\) so that the resulting statement is true. 5 ________ |-2|

Step-by-Step Solution

Answer

Answer: The correct inequality symbol to place between 5 and |-2| is greater than (>).

Step 1: Find the absolute value of -2

To find the absolute value of -2, we need to determine how far the number is from zero. The absolute value of -2 is 2 because it is two units away from zero. Thus, |-2| = 2.

Step 2: Compare 5 and |-2|

Now that we know the absolute value of -2 is 2, we can compare 5 and 2. Since 5 is greater than 2, we can write this relationship using the greater than symbol (>): 5 > 2.

Step 3: Replace |-2| with 2 and write the final inequality

In our comparison, we found that 5 is greater than the absolute value of -2, which is 2. So our inequality should be: 5 > |-2|. We can write this as: 5 > 2.

Key Concepts

Inequalities

Inequalities are mathematical statements that show the relative size or order of two values. They tell us whether a number is greater than, less than, or equal to another number. Inequalities use symbols to express these relationships, such as:

• Less than: \(<\)
• Greater than: \(>\)
• Less than or equal to: \(\leq\)
• Greater than or equal to: \(\geq\)

By comparing numbers or expressions, inequalities help us understand how these values relate to each other. For example, if we have two numbers, 5 and 2, and we know 5 is greater than 2, we can write this relationship as \(5 > 2\). Inequalities are used in various mathematical contexts such as equations, functions, and real-world problems to define constraints or conditions.

Comparing Numbers

Comparing numbers is essential for understanding their quantitative relationships. When we compare numbers, we determine if one number is larger, smaller, or equal to another. To compare numbers effectively, consider the following steps:

• Look at the numbers individually and determine their sizes. For instance, when comparing 5 and 2, we see that 5 is larger than 2.
• Use mathematical symbols like \(>\), \(<\), or \(=\) to express the comparison. Hence, we say \(5 > 2\) when 5 is larger than 2.

Sometimes, the numbers you compare may be part of an expression, like considering the absolute value of a number. Absolute value represents the distance from zero, so it is always a non-negative number.Additionally, comparing numbers involves understanding both the numerical size and considering properties like absolute values, which may alter how we view these numbers. This is especially true when using absolute values in equations or inequalities.

Mathematical Symbols

Mathematical symbols are essential tools for expressing numbers and their relationships in a concise manner. In our exercise, the focus is on symbols related to inequality:

• Greater than (>): This symbol indicates that the number or expression on the left side is larger than the one on the right.
• Less than (<): Indicates that the left-side number is smaller than the right-side number.
• Equal to (=): Expresses that two numbers or expressions are the same in value.

Using these symbols appropriately allows mathematicians and students to quickly and clearly show how numbers relate to each other. For example, the problem from the exercise uses the greater than symbol (>): - 5 > |-2| Here, the expression indicates that 5 is greater than the absolute value of -2, which is 2. Understanding how to use these symbols in context is crucial to solving equations and expressing relationships clearly in mathematics.