Problem 45 - College Algebra Essentials

Question

Determine whether each statement in Exercises 43–50 is true or false. $$4 \geq-7$$

Step-by-Step Solution

Answer

The statement '4 ≥ -7' is true.

Step 1: Understanding the inequality sign.

The symbol '≥' is a type of inequality and it means 'greater than or equal to'. When you see 'A ≥ B', it means that A is either greater than B or equal to B.

Step 2: Comparing the numbers.

In this case, we need to evaluate if '4 is greater than or equal to -7'. On the number line, 4 is to the right of -7, which means it is greater. Therefore, the statement '4 ≥ -7' is true.

Key Concepts

Greater Than or Equal To

The concept of "greater than or equal to" is a fundamental principle in mathematics often represented by the symbol "≥". This symbol is used in inequalities to compare two values or expressions. When you see a statement like "A ≥ B", it suggests two possibilities:

• A is greater than B, meaning A is to the right of B on a number line.
• A is equal to B, meaning they are the same number.

In simple terms, this sign tells us that A cannot be less than B. This concept helps us understand ordering and comparing quantities in math, which is critical for solving many mathematical problems.

Number Line

A number line is an essential tool in mathematics used to visually represent numbers and their relationships. It is a straight line with numbers placed at equal intervals along its length. The left side of the number line usually represents smaller numbers, and as you move right, the numbers increase. When dealing with inequalities such as "4 ≥ -7", a number line can help us visually see that 4 is indeed greater since it appears to the right of -7.

Using a number line can simplify comparisons and make it easier to grasp how numbers relate to each other. It provides a clear visual context where you can easily see which number is larger or smaller, assisting in determining the truth of inequality statements.

True or False Statements

When we talk about true or false statements in mathematics, we are essentially determining the validity of an expression or comparison. To declare a statement as true means that the information or condition given is correct. For instance, in our example, "4 ≥ -7", the comparison of these numbers results in a true statement because 4 is actually greater than -7.

Statements are fundamental in math to assess factual accuracy. To evaluate the truth of any inequality:

• Look at the given numbers or expressions.
• Use mathematical principles, such as number lines or arithmetic comparisons, to judge their relationship.
• State whether the initial assertion holds or not.

By understanding these steps, you can systematically determine the truthfulness of various mathematical statements.