Rational Symbols in Math

• Rational symbols are used to compare numbers and mathematical expressions
• These symbols indicate the relative size or order of the values being compared

Common Rational Symbols

• = (equal to): Indicates that two values are the same
• ≠ (not equal to): Indicates that two values are different

• (greater than): Indicates that the left value is larger than the right value

• < (less than): Indicates that the left value is smaller than the right value
• ≥ (greater than or equal to): Indicates that the left value is larger than or the same as the right value
• ≤ (less than or equal to): Indicates that the left value is smaller than or the same as the right value

Using the "=" Symbol

• The "=" symbol asserts equality between two expressions
• Example: 5 = 5 (5 is equal to 5)
• Example: x + 2 = 7 (x + 2 is equal to 7, implying x = 5)
• If both sides of the "=" symbol evaluate to the same value, the statement is true

Using the "≠" Symbol

• The "≠" symbol asserts inequality between two expressions
• Example: 3 ≠ 4 (3 is not equal to 4)
• Example: x ≠ 5 (x is not equal to 5)
• The statement is true if both sides of the "≠" symbol do not evaluate to the same value

Using the ">" Symbol

• The ">" symbol indicates that the left value is greater than the right value
• Example: 7 > 3 (7 is greater than 3)
• Example: x > 10 (x is greater than 10, meaning x can be any number larger than 10)
• The statement is true if the left side has a larger numerical value than the right side

Using the "<" Symbol

• The "<" symbol indicates that the left value is less than the right value
• Example: 2 < 6 (2 is less than 6)
• Example: y < 0 (y is less than 0, meaning y is any negative number)
• The statement is true if the left side has a smaller numerical value than the right side

Using the "≥" Symbol

• The "≥" symbol indicates that the left value is greater than or equal to the right value
• Example: 5 ≥ 5 (5 is greater than or equal to 5)
• Example: x ≥ 1 (x is greater than or equal to 1, meaning x can be 1 or any number larger than 1)
• The statement is true if the left side has a numerical value greater than or the same as the right side

Using the "≤" Symbol

• The "≤" symbol indicates that the left value is less than or equal to the right value
• Example: 4 ≤ 4 (4 is less than or equal to 4)
• Example: y ≤ -2 (y is less than or equal to -2, meaning y can be -2 or any number smaller than -2)
• The statement is true if the left side has a numerical value less than or the same as the right side

Comparing Negative Numbers

• When comparing negative numbers, the number closer to zero is greater
• Example: -2 > -5 (-2 is greater than -5)
• Example: -10 < -1 (-10 is less than -1)
• Use the number line to visualize the order: numbers increase from left to right

Comparing Fractions

• To compare fractions, find a common denominator
• Convert each fraction to an equivalent fraction with the common denominator
• Compare the numerators: the fraction with the larger numerator is greater
• Example: Comparing 1/2 and 2/5: Convert to 5/10 and 4/10 respectively. 5/10 > 4/10, so 1/2 > 2/5
• Alternatively, convert fractions to decimals and compare the decimal values

Comparing Decimals

• Compare the whole number parts first
• If whole number parts are equal, compare the tenths place, then the hundredths place, and so on
• Example: 3.14 > 3.11 (comparing the hundredths place)
• Example: 0.5 < 0.75 (comparing the tenths place)

Comparing Real Numbers

• Real numbers include rational and irrational numbers
• Rational numbers can be expressed as a fraction
• Irrational numbers (like π or √2) cannot be exactly expressed as a fraction
• Use decimal approximations to compare irrational numbers
• Example: π ≈ 3.14159, √10 ≈ 3.16228. Thus, √10 > π

Transitive Property

• If a > b and b > c, then a > c
• If a < b and b < c, then a < c
• This property applies to =, ≥, and ≤ as well

Comparison with Variables

• When comparing expressions with variables, determine the possible range of values
• Example: If x > 5, then x + 2 > 7
• Use algebraic manipulation to simplify and compare expressions
• Example: If x + 3 < 8, then x < 5

Absolute Value in Comparisons

• Absolute value |x| represents the distance of x from 0
• |-3| = 3 and |3| = 3
• When comparing absolute values, consider both positive and negative values inside the absolute value
• Example: |5| > |-2| (5 > 2)
• Example: |-4| < |-6| (4 < 6)

Comparing Numbers on a Number Line

• Numbers to the right on the number line are greater
• Numbers to the left on the number line are smaller
• Visualizing numbers on a number line can aid in comparing them, especially negative numbers

Complex comparisons

• Multiple comparisons can be combined for more complex boolean logic
• For example, a < x < b means x is greater than a but less than b
• x >= a and x <= b means x is greater or equal to a and less than or equal to b, which means it is within [a,b] inclusive