Problem 60 - Elementary and Intermediate Algebra

Question

Classify each inequality as either true or false. $$-5 \leq 7$$

Step-by-Step Solution

Answer

True

Step 1: Understand the Inequality

The given inequality is \( -5 \leq 7 \). The symbol \( \leq \) means 'less than or equal to'. Hence, we need to check if -5 is indeed less than or equal to 7.

Step 2: Compare the Values

Compare the two numbers in the inequality: -5 and 7. Answer the question: Is -5 less than or equal to 7?

Step 3: Determine the Truth

Since -5 is less than 7, the statement \( -5 \leq 7 \) is true.

Key Concepts

inequality symbols

In algebra, inequality symbols are essential in comparing values. These symbols include:

• < (less than)
• > (greater than)
• \(egless than or equal to)
• \(eg>\) (greater than or equal to)
• \(eq\) (not equal to).

Each symbol allows us to express different relationships between two values. Knowing these helps in understanding and solving inequalities. For instance, in the inequality -5 \(eg\leq\) 7, the \(egegegegegegeg\) symbol indicates that we are checking if -5 is less than or equal to 7.

comparing values

Comparing values in an inequality involves looking at the relative sizes of the numbers. In the example -5 \(eg\leq\) 7, we compare -5 and 7. To do this, visualize their positions on a number line:

• -5 is to the left of 0
• 7 is to the right of 0.

Numbers to the left on the number line are smaller than those to the right. Therefore, -5 is indeed less than 7, satisfying the condition expressed by the inequality symbol. This visual tool can simplify comparing values.

truth value of inequalities

The truth value of an inequality tells us whether the statement is true or false. To determine this, follow these steps:

• Identify the inequality symbol
• Compare the two values
• Decide if the relationship holds true.

In the example -5 \(eg\leq\) 7, we've identified the symbol \(\leq\), meaning 'less than or equal to'.

Comparing -5 and 7 confirms that -5 is less than 7. Hence, the inequality holds true, and we say the truth value of this inequality is true.