Problem 14 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The inequality is true.

Step 1: Identify the Inequality

The given inequality is \(8 \leq 9\). This means we need to determine if 8 is less than or equal to 9.

Step 2: Understand the Symbols

The symbol \(\leq\) means 'less than or equal to.' The inequality \(8 \leq 9\) states that 8 is either less than or equal to 9.

Step 3: Evaluate the Numeric Relationship

Compare the numbers 8 and 9. Since 8 is indeed less than 9, the inequality holds true.

Step 4: Determine Truth Value

Since 8 is less than 9, this supports the truth of the statement \(8 \leq 9\). Thus, the inequality \(8 \leq 9\) is true.

Key Concepts

Understanding 'Less Than or Equal To'

In mathematics, the symbol \(\leq\) represents the phrase "less than or equal to." This symbol is used in inequalities to compare two values, indicating that the first number is either less than or actually equal to the second number. For instance, the expression \(8 \leq 9\) means 8 can be either less than or equal to 9. This concept is quite common in mathematics, especially when dealing with number comparisons, functions, and real-world situations like budgeting or measurement. Remember, when you see \(\leq\), check if the first number is smaller or equal to the second. If either condition is true, the whole inequality is true.
Here are a few quick examples:

• \(4 \leq 5\) is true because 4 is less than 5.
• \(7 \leq 7\) is true because 7 equals 7.
• \(10 \leq 8\) is false because 10 is neither less than nor equal to 8.

Understanding this symbol helps in accurately solving and evaluating inequalities.

Truth Value of an Inequality

Evaluating the truth value of an inequality is essentially determining whether the statement is true or false. The truth value tells us if the assertion made by the inequality holds with the given numbers. For example, in the inequality \(8 \leq 9\), we ask: "Is 8 less than or equal to 9?" If the answer is yes, then the truth value is true, otherwise false.
To determine if a statement like \(x \leq y\) is true:

• Confirm whether \(x\) is less than \(y\), which makes it true.
• If \(x\) and \(y\) are equal, it still means the inequality is true.
• Only when \(x\) is greater than \(y\), the statement becomes false.

Mastering how to find the truth value of an inequality is important because it helps in solving many math problems accurately, ensuring the analysis aligns with logical truth.

How to Evaluate an Inequality

Evaluating an inequality involves comparing the given values and deciding if the inequality holds. Unlike equations that signify equality, inequalities tell us about relative sizes such as being lesser, greater, or equal. To evaluate an inequality successfully, you need to:

• Identify the inequality. For example, \(8 \leq 9\) where \(8\) is to be compared with \(9\).
• Understand the inequality symbol. In this case, \(\leq\) means less than or equal to.
• Apply logical comparison of the numerical values, seeing if the first number follows the relation with the second as the inequality suggests.
• Decide the truth value. Since 8 is less than 9, the inequality \(8 \leq 9\) holds true.

By following these steps, you make sure every part of the inequality is considered and that your final judgment is correct. This skill is useful in both academic work and everyday reasoning, such as comparing prices or distances.