Problem 16 - Algebra and Trigonometry

Question

\(9-16\) State whether each inequality is true or false. $$ 8 \leq 8 $$

Step-by-Step Solution

Answer

The inequality \(8 \leq 8\) is true.

Step 1: Understand the Inequality

The inequality given is \(8 \leq 8\). It uses the 'less than or equal to' symbol (\(\leq\)), which means the left side should be either less than or equal to the right side.

Step 2: Substitute Values

The inequality compares 8 to 8. Replace both sides with the numerical values to see if the statement is true or false.

Step 3: Evaluate Comparison

Check if 8 is less than or equal to 8. Since 8 is equal to 8, the statement holds true under the 'less than or equal to' condition.

Step 4: Conclusion

Based on the evaluation, determine whether the inequality is true or false. Since 8 equals 8, the inequality \(8 \leq 8\) is true.

Key Concepts

Understanding 'Less Than or Equal To'

The concept of 'less than or equal to', symbolized by \(\leq\), is a fundamental part of understanding inequalities in mathematics. It signifies that a number on the left side must either be smaller than or exactly the same as the number on the right side of the inequality.
For example, if we have \(x \leq y\), it means that \(x\) can be less than \(y\), or \(x\) can be exactly equal to \(y\). This dual condition is crucial.

• If \(x\) is 5 and \(y\) is 7, then \(5 \leq 7\) is true because 5 is less than 7.
• If \(x\) is 7 and \(y\) is 7, then \(7 \leq 7\) is also true because 7 equals 7.

Thus, 'less than or equal to' allows flexibility—any situation where the left number is either lesser or exactly equal meets the condition.

Evaluating Inequalities

Evaluating inequalities involves understanding the relationship between two values or expressions to determine their comparative status. Here’s a simple way to do it:
First, break down the inequality to see the numbers or expressions on each side, just like the inequality \(8 \leq 8\).

• Identify the numbers or variables: In the example, we identify 8 on both sides of the inequality.
• Compare the values: Determine if the number on the left is less than or equals the number on the right. Here, 8 is equal to 8.

Once you have identified and compared, you are ready to decide if the inequality is true or false, which leads to understanding of evaluating differences and relationships between mathematical expressions.

True or False Statements in Inequalities

Inequalities can often be validated by determining the truth of their statements. A true statement correctly describes the relationship between quantities, while a false statement does not.
To check if an inequality is true or false, follow these steps:

• Identify the expressions on both sides of the inequality sign, which in this case are both 8 in \(8 \leq 8\).
• Evaluate whether the condition of the inequality is met. Is the left side less than or equal to the right side?
• In \(8 \leq 8\), since 8 equals 8, the statement is true.

Being able to determine the truth value of a statement is an essential skill, helping build a deeper understanding of both simple and complex mathematical concepts.