Problem 13 - Algebra 1: Concepts and Skills

Question

Write the numbers in increasing order. \(-7,-9,2, \frac{5}{4},-\frac{1}{5}\)

Step-by-Step Solution

Answer

So the numbers arranged from smallest to largest are \( -9, -7, -\frac{1}{5}, \frac{5}{4}, 2 \).

Step 1: Identifying negative numbers

Among the given numbers, \( -7, -9, \) and \( -\frac{1}{5} \) are negative.

Step 2: Arrange the negative numbers

Among the negative numbers, the number closer to zero is greater. So, we have \( -9 < -7 < -\frac{1}{5} \).

Step 3: Identify positive numbers

Among the given numbers, \( 2 \) and \( \frac{5}{4} \) are positive.

Step 4: Change fraction to decimal

\(\frac{5}{4} \) can be changed to \( 1.25 \) by performing the division. This makes it easier to compare with other numbers.

Step 5: Arrange the positive numbers

Among the positive numbers, the number closer to zero is smaller. So, we have \( \frac{5}{4} < 2 \).

Step 6: Combine positive and negative numbers

Putting the negative numbers first and then the positive numbers we get \( -9 < -7 < -\frac{1}{5} < \frac{5}{4} < 2 \) as the final increasing order.

Step 7: Substitute back the originals

Substituting back the originals where required. We get the final order as \( -9 < -7 < -\frac{1}{5} < \frac{5}{4} < 2 \).

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as fractions. This means they consist of a numerator and a denominator. For example, the fraction \(\frac{5}{4}\) is a rational number because it can be divided into the decimal 1.25.

• A rational number can be positive, such as 1.25, or negative, such as \(-\frac{1}{5}\).
• They include whole numbers because integers like 2 can be written as fractions: \(\frac{2}{1}\).

Understanding rational numbers is crucial because it allows you to work with all types of numbers commonly seen in everyday math, including fractions and decimals. Rational numbers make ordering tasks easier because they provide a clear scale from negative numbers through to positive numbers.

Positive and Negative Numbers

Understanding the concept of positive and negative numbers helps us make sense of the order and position on the number line. Negative numbers are found to the left of zero, and positive numbers are to the right.

• Negative numbers, like -9, -7, and -\(\frac{1}{5}\), are lower in value than any positive number.
• The further left you go on the number line, the smaller the value. Hence, -9 is smaller than -7.
• Positive numbers, such as 2 and 1.25, are greater than zero and each negative number.

Ordering numbers requires understanding which of these groups (positive or negative) a number belongs to. Then, you can use their distance from zero to determine their order.

Comparison of Decimals

Comparing decimals often involves few steps: converting fractions into decimals, ensuring uniformity, and then examining the numbers. This makes it easier to visualize and compare them.

• For instance, \(\frac{5}{4}\) becomes 1.25, which can be directly compared to whole numbers or other decimals.
• Decide which number is closer to zero to determine its placement.
• Decimals between 0 and 1, like -0.2, are negative but closer to zero than decimals like -7 or -9.

Thus, in comparison, -0.2 is greater than -9 or -7. Conversion into decimals streamlines the process by aligning numbers into an easily comparable form.