Problem 43 - College Algebra

Question

Determine whether statement is true or false. \(-13 \leq-2\)

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understand the inequality

The inequality \(\leq\) stands for 'less than or equal to'. This inequality symbol is comparing two numbers, where the number on the left could be less than or equal to the number on the right.

Step 2: Understand the given numbers

In the order of numbers on a number line, any negative number is less than any positive number, and absolute value has an effect on the placement of negative numbers. The higher the absolute value of a negative number, the smaller that number is. Here, \(-13\) and \(-2\) are being compared, while both are negative numbers, \(-13\) has a greater absolute value which means it's smaller than \(-2\).

Step 3: Answer the question

So, the statement \(-13\) is less than \(-2\) is true because \(-13\) is indeed less than \(-2\). Thus, \(-13 \leq -2\) is a true statement.

Key Concepts

Negative Numbers

Negative numbers are less than zero and are represented with a minus sign preceding the number. They can be thought of as a reflection of their positive counterparts but on the opposite side of zero on the number line. In the context of inequalities:

• Negative numbers are always smaller than positive numbers, as they lie to the left on the number line.
• Comparing two negative numbers works similarly to positive numbers, but it's important to focus on their absolute values. The greater the absolute value of a negative number, the smaller the number itself.

For example, egin{align*} -5 < -3 ext{because } 5 > 3. ewline ext{So even though 5 is larger, } -5 ext{ is smaller than } -3. ewline ext{This concept is key whenever dealing with negative numbers in inequalities.} Understanding this allows us to correctly interpret and solve inequalities involving negatives.

Number Line

A number line is a visual tool used to understand the placement and order of numbers. It helps illustrate the relationship between numbers, whether they are positive or negative.

• A number line is typically drawn horizontally, with zero in the middle, positive numbers to the right, and negative numbers to the left.
• The further left you go, the smaller the numbers become.
• Number lines are essential for visualizing and solving problems related to ordering and comparing numbers.

To compare numbers on a number line, you simply look at their relative positions:

• Any number to the left is smaller than any number to the right.
• This makes it easy to visualize inequalities and determine which statements are true or false.

Using a number line helps reinforce the concept that egin{align*} -13 < -2, ewline ext{as -13 is further left on the number line, indicating it's smaller.} Understanding number lines makes comparing numbers much more intuitive.

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, without considering direction. It is always a non-negative number.

• The absolute value of any number, whether positive or negative, is positive.
• For example, the absolute value of both -5 ext{ and } 5 ext{ is } 5, written as |−5| = 5 and |5| = 5.
• Absolute value is used primarily when comparing sizes of numbers without adhering to their direction.

In inequalities involving negative numbers, understanding absolute value is crucial.

• Though -13 ext{ has an absolute value of } 13 ext{ and } -2 ext{ has } 2, -13 is considered smaller because it lies further left on the number line.

Recognizing how absolute value affects the placement of negative numbers ensures accurate comparison and true statements in inequalities.