Problem 143 - Prealgebra and Introductory Algebra

Question

Place the correct symbol, \(<\) or \(>,\) between the numbers. $$-\frac{3}{4} \quad-0.7$$

Step-by-Step Solution

Answer

The correct symbol to place between -3/4 and -0.7 is \(<\), yielding the relationship -3/4 < -0.7.

Step 1: Convert Fraction to Decimal

The first step would be to convert the given fraction, -3/4, into a decimal. Since a fraction represents division, we simply divide the numerator by the denominator. That is, -3 divided by 4. The result of this division, to one decimal place, would be -0.75.

Step 2: Compare the Decimals

The next step is to compare the two decimals. Here, we have -0.75 and -0.7. Decimal numbers have a property that farther to the left means a larger value if comparing two numbers. Therefore, because -0.75 is farther to the left than -0.7, it implies that -0.7 is greater than -0.75.

Step 3: Insert the Appropriate Symbol

We have compared the two decimals and found that -0.7 is greater than -0.75. Therefore, when we insert the appropriate symbol between the two numbers (in their original given formats), we obtain: -3/4 < -0.7.

Key Concepts

Fractions and Decimals

Fractions and decimals are two ways to represent numbers, especially those that are not whole. A fraction such as \(-\frac{3}{4}\) consists of a numerator (the top number) and a denominator (the bottom number). When you see a fraction, think of it as dividing the numerator by the denominator. In this case, \(-3 \div 4\) tells us how many times 4 goes into 3, resulting in the decimal \(-0.75\).

Decimals are another way to express numbers, often making them easier to compare and calculate with, as they align with our base-ten numbering system. Converting fractions to decimals can simplify understanding, especially when dealing with smaller or negative numbers.

Remember, converting fractions into decimals is as simple as performing division. This process helps make comparisons between numbers more straightforward.

Number Line

A number line is a visual tool that helps us understand the value of numbers by arranging them in a line according to size. On a typical number line:

• Numbers to the right are greater than numbers to the left.
• Positive numbers are located on the right side of zero.
• Negative numbers are found on the left side of zero.

When comparing numbers like \(-0.75\) and \(-0.7\), a number line provides a clear picture of their positions. Since \(-0.75\) is farther to the left of \(-0.7\) on the number line, \(-0.7\) is closer to zero and therefore larger.

Using a number line is especially helpful for visual learners, helping them grasp abstract concepts like negative values and their comparative sizes. It's a simple yet powerful way to see which of two numbers is larger or smaller.

Inequalities

Inequalities are mathematical expressions used to compare two values. They tell us if one value is greater than, less than, or equal to another. The symbols used for inequalities are:

• \(<\) meaning "less than."
• \(>\) meaning "greater than."

When we say \(-\frac{3}{4} < -0.7\), we are using an inequality to state that \(-\frac{3}{4}\) is less than \(-0.7\). This was determined by converting \(-\frac{3}{4}\) into \(-0.75\), allowing a straightforward comparison with \(-0.7\).

Inequalities are crucial because they help express relationships between numbers and are used extensively in equations and problem solving. Always pay close attention to the direction of the inequality, as it indicates the relative size of the numbers involved.