Problem 66 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each inequality is true or false. $$-14 \leq-14$$

Step-by-Step Solution

Answer

The given inequality -14 ≤ -14 is true. Because -14 is indeed equal to -14.

Step 1: Analyze the inequality

The inequality given is -14 ≤ -14. This inequality states that the number -14 is less than or equal to -14.

Step 2: Direct comparison

Comparing the numbers, -14 is not less than -14, but they are equal. Hence inequality holds true iff the comparison were inclusive of equality.

Key Concepts

Comparison of numbers

Comparing numbers is one of the basic skills in math that helps us understand the relationship between different values. In the case of the inequality \(-14 \leq -14\), we are comparing the number \(-14\) with itself. When comparing numbers, we look at:

• Magnitude: The size of the numbers, which tells us whether one number is greater, lesser, or equal to another.
• Sign: Whether the numbers are positive or negative.

In our exercise, both numbers are the same, meaning their magnitude and sign are identical. We often denote this scenario with a double-ended inequality symbol, such as "<=" or ">=" indicating that the numbers can be equivalent.

True or false inequalities

Inequalities help us understand whether one expression is bigger, smaller, or the same as another. An inequality can either be true or false:

• True: If the relationship described by the inequality holds.
• False: If it does not.

For the inequality \(-14 \leq -14\), we check these two possibilities:- "Less than": \(-14\) is not less than \(-14\), so this part is false.- "Equal to": \(-14\) equals \(-14\), making this part true.
Since the inequality uses \("\leq"\), which includes "equal to", this inequality is true.

Inclusive comparison

Inclusive comparison is a type of comparison where equality is considered alongside the inequality. In symbols like \("\leq"\) and \(">="\), we read them as "less than or equal to" and "greater than or equal to", respectively. The comparison is inclusive because:

• "Less than or equal to" means the first number can be smaller or exactly the same as the second number.
• "Greater than or equal to" means the first number can be larger or exactly the same as the second number.

In the exercise with \(-14 \leq -14\), the term "less than or equal" includes the possibility of the numbers being identical, which is why the statement is true. This inclusive aspect is crucial in understanding and correctly solving inequalities.