Problem 74 - Beginning and Intermediate Algebra

Question

Select the lesser of the two given numbers. \(|7-2|,|8-1|\)

Step-by-Step Solution

Answer

5

Step 1: Calculate the Absolute Value of the First Expression

First, compute the absolute value of the expression |7-2|. The absolute value of a number is its distance from zero on the number line, without considering the direction.

Step 2: Simplify the First Expression

Simplify inside the absolute value to get |7-2| = |5|. Since 5 is already positive, this becomes 5.

Step 3: Calculate the Absolute Value of the Second Expression

Next, compute the absolute value of the expression |8-1|. Again, absolute value represents distance from zero.

Step 4: Simplify the Second Expression

Simplify inside the absolute value to get |8-1| = |7|. Since 7 is already positive, this becomes 7.

Step 5: Compare the Two Values and Select the Lesser

Compare the two results, 5 and 7. The lesser of the two numbers is 5.

Key Concepts

Understanding Absolute Value

Absolute value is a fundamental concept in mathematics. It represents the distance of a number from zero on the number line. This distance is always considered as a positive quantity.

For example, the absolute value of both -3 and +3 is 3 because both are three units away from zero. The notation used is vertical bars around the number. So, \(|-3| = 3\) and \(|3| = 3\).

When you work with absolute values in an equation or expression, simplify the inside first, and then apply the absolute value. In our exercise, we simplify \(|7-2|\) and \(|8-1|\) first before applying the absolute value.

Integer Operations Made Simple

Integer operations include basic arithmetic functions like addition, subtraction, multiplication, and division, involving whole numbers. For the given exercise, we focus on subtraction before applying absolute value.

Here's a quick tip:

• Always perform operations inside any absolute value bars first.

For example, in \(7-2\), you perform the subtraction to get 5. Similarly, in \(8-1\), you get 7. These are integer operations that simplify your problem-solving process.

Once you have simplified the integers, apply the absolute value if required. In our exercise, after subtracting inside the absolute bars, we get positive numbers, so taking absolute value doesn't change them.

Number Comparison Key Points

Comparing numbers is essential in many math problems. It's about determining which number is greater, lesser, or if they are equal.

In the given exercise, we compare the results of the absolute values: 5 and 7. Since 5 is less than 7, 5 is our answer.

Always remember:

• Simplify the numbers first.
• Apply absolute values if necessary.
• Perform the comparison: check which number is smaller, greater, or if they are equal.

Practicing these steps will help you efficiently compare numbers and solve similar exercises effortlessly.