Problem 35 - Elementary Algebra

Question

For the pairs of real numbers shown in the following problems, write the appropriate relation symbol \((<,>,=)\) in place of the \(*\) $$-\frac{1}{4} *-\frac{3}{4}$$

Step-by-Step Solution

Answer

Question: Compare the two numbers, $$-\frac{1}{4}$$ and $$-\frac{3}{4}$$, and determine which number is greater, less, or if they are equal. Write the appropriate symbol between the two numbers. Answer: $$-\frac{1}{4} > -\frac{3}{4}$$

Step 1: Compare the numerators

Since both of the numbers in question have the same denominator, we can directly compare their numerators. In this case, we have -1 and -3.

Step 2: Determine the appropriate relation symbol

We can see that -1 is greater than -3, because on the number line, -1 lies to the right of -3. Therefore, the appropriate symbol is greater than (>). Replace the * symbol with >.

Step 3: Final Answer

The final expression will be: $$-\frac{1}{4} > -\frac{3}{4}$$

Key Concepts

Real Numbers

Real numbers are all numbers that exist on the number line. They include all rational numbers, like fractions and integers, as well as irrational numbers, like \( \pi \) and \( \sqrt{2} \). Real numbers can either be positive, negative, or zero. This encompassing set allows us to perform all basic arithmetic operations and engage in complex comparisons. Real numbers are crucial in everyday mathematics because they provide a complete picture of what we understand as numbers.

• Every real number can be plotted on the number line.
• Real numbers that are not fractions or decimals include integers such as \( -2, 0, 3 \).
• The set of real numbers is denoted by \( \mathbb{R} \).

Fractions

Fractions represent a part of a whole and are written as \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator. In comparing fractions with the same denominator, the fraction with the larger numerator is the larger fraction. Negative fractions are less than positive fractions, and among negative fractions, the one closer to zero is less negative, thus larger.

• When denominators are equal, compare numerators directly to determine which fraction is larger.
• Fractions can be used in addition, subtraction, multiplication, and division.
• If a numerator is negative, the entire fraction is negative.

Order of Numbers

Ordering numbers involves arranging them from the smallest to the largest or vice versa. To compare real numbers, especially fractions, consider their position on a number line. For negative numbers, the closer to zero, the larger the number will be. In inequalities, this is reflected by the use of symbols like \( < \) for less than, \( > \) for greater than, and \( = \) for equal to.

• Numbers on the right side of the number line are larger than those on the left.
• Use inequality symbols to express the order of numbers.
• When comparing negative numbers, consider that the "less negative" number is the larger one.