Problem 58 - Elementary and Intermediate Algebra

Question

Classify each inequality as either true or false. $$ 5 \leq-5 $$

Step-by-Step Solution

Answer

False

Step 1: Understanding the Symbols

The symbol \(\backslashleq\) means 'less than or equal to'. This means that the statement \(\backslash( 5 \backslashleq -5 \backslash)\) is claiming that 5 is less than or equal to -5.

Step 2: Compare the Numbers

Compare the two numbers: 5 and -5. We know that 5 is greater than -5 because positive numbers are always greater than negative numbers.

Step 3: Determine the Truth Value

Since 5 is greater than -5, the statement \(\backslash( 5 \backslashleq -5 \backslash)\) is false. 5 is neither less than -5 nor equal to -5.

Key Concepts

inequality symbols

In algebra, we use various symbols to express inequalities. These symbols help us compare two values.
One such symbol is \( \leq \), which stands for 'less than or equal to'.
It's crucial to understand what each symbol means to accurately interpret and solve inequality problems. Let's look at the common inequality symbols and their meanings:

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

By knowing these, you can easily understand problem statements involving inequality and will be able to compare the numbers correctly.

comparing numbers

Now that we understand the inequality symbols, let's dive into comparing numbers.
In the given exercise, we need to decide if 5 is less than or equal to -5.
To do this, we simply compare the two values to see which one is greater. In our number system:

• Positive numbers are always greater than negative numbers
• Zero is greater than any negative number but less than any positive number
• Within the same sign category (both positive or both negative), bigger absolute values mean bigger numbers for positives but smaller for negatives

Applying this, 5 is clearly greater than -5, so saying that 5 is less than or equal to -5 is incorrect.
This comparison helps us determine the truth of the inequality.

truth value of inequalities

Understanding the truth value of inequalities is essential to solving algebra problems.
The truth value indicates whether a certain statement about two numbers holds true or not.
In the context of inequalities, the truth value is either 'true' or 'false'.
For the inequality \( 5 \leq -5 \), after determining that 5 is greater than -5, we conclude that the statement is false.
This is because the claim that 5 is less than or equal to -5 does not hold under any circumstances.
Checking the truth value involves these steps:

• Understand the inequality symbol
• Compare the numbers
• Apply logical reasoning to see if the statement matches the real-world relationship of those numbers

By rigorously following these steps, you can solve any inequality and correctly identify its truth value.