Mathematical Symbols

Equal To (=)

• The equals sign (=) asserts that two numbers have the same value.
• Example: 5 = 5, meaning the number 5 is equal to itself.
• In algebraic terms: if a = b, then 'a' and 'b' represent the same quantity.

Not Equal To (≠)

• The not equal sign (≠) indicates that two numbers do not have the same value.
• Example: 7 ≠ 9, meaning the number 7 is not equal to the number 9.
• This symbol is the negation of the equals sign.

Greater Than (>)

• The greater than sign (>) shows that one number has a larger value than another.
• Example: 12 > 4, meaning the number 12 is greater than the number 4.
• On a number line, the greater number is located to the right of the smaller number.

Less Than (<)

• The less than sign (<) shows that one number has a smaller value than another.
• Example: 3 < 8, meaning the number 3 is less than the number 8.
• On a number line, the smaller number is located to the left of the larger number.

Greater Than or Equal To (≥)

• The greater than or equal to sign (≥) indicates that one number is either larger than or equal to another.
• Example: x ≥ 10, meaning x can be 10 or any number larger than 10.
• This combines the concepts of 'greater than' and 'equal to'.

Less Than or Equal To (≤)

• The less than or equal to sign (≤) indicates that one number is either smaller than or equal to another.
• Example: y ≤ 5, meaning y can be 5 or any number smaller than 5.
• This combines the concepts of 'less than' and 'equal to'.

Number Line Representation

• Numbers can be visually compared using a number line.
• Numbers to the right are always greater than numbers to the left.
• This representation helps in understanding inequalities and relative values.

Absolute Value

• Absolute value, denoted as |x|, represents the distance of a number x from zero on the number line.
• Absolute value is always non-negative.
• Example: |-5| = 5 and |5| = 5.

Comparing Absolute Values

• When comparing absolute values, consider the distance from zero, not the sign of the number.
• For instance, |-7| > |3| because 7 (the absolute value of -7) is greater than 3 (the absolute value of 3).

Comparing Negative Numbers

• When comparing negative numbers, the number closer to zero is greater.
• Example: -2 > -6, because -2 is closer to zero than -6 on the number line.
• The further a negative number is from zero, the smaller its value.

Transitive Property

• If a > b and b > c, then a > c. This is the transitive property.
• This property applies to all inequality symbols (> , < , ≥, ≤).

Addition and Subtraction Properties

• Adding the same number to both sides of an inequality preserves the inequality.
• If a > b, then a + c > b + c.
• Subtracting the same number from both sides also preserves the inequality.
• If a > b, then a - c > b - c.

Multiplication and Division Properties

• Multiplying or dividing both sides of an inequality by a positive number preserves the inequality.
• If a > b and c > 0, then a * c > b * c and a / c > b / c.
• Multiplying or dividing both sides of an inequality by a negative number reverses the inequality.
• If a > b and c < 0, then a * c < b * c and a / c < b / c.

Compound Inequalities

• Compound inequalities combine two or more inequalities.
• 'And' inequalities (e.g., a < x and x < b) represent an intersection.
• 'Or' inequalities (e.g., x < a or x > b) represent a union.

Interval Notation

• Interval notation is used to represent a range of numbers.
• Parentheses ( ) indicate exclusion of endpoints.
• Brackets [ ] indicate inclusion of endpoints.
• Examples: (a, b) means a < x < b, [a, b] means a ≤ x ≤ b.

Infinity Symbol

• Infinity (∞) represents a quantity without bound.
• Negative infinity (-∞) represents a quantity negatively without bound.
• Infinity is always enclosed in parentheses in interval notation because it is not a specific number.

Comparing Fractions

• To compare fractions, find a common denominator.
• Once the denominators are the same, compare the numerators.
• The fraction with the larger numerator is the greater fraction.

Comparing Decimals

• To compare decimals, align the decimal points.
• Compare the digits from left to right.
• The decimal with the larger digit in the highest place value is the greater decimal.

Comparing Percentages

• Percentages can be compared directly once they are in the same context.
• Convert percentages to decimals or fractions for comparison with other types of numbers.

Comparing Radicals

• To compare radicals, simplify them if possible.
• If the radicals have the same index, compare the radicands (the values inside the radical).
• If the indices are different, rationalize or convert to a common index.

Scientific Notation

• Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.
• To compare numbers in scientific notation, compare the exponents first.
• If the exponents are the same, compare the numbers between 1 and 10.

Ordering Numbers

• Ordering numbers involves arranging them from least to greatest or greatest to least.
• Use comparison symbols to determine the relative order of the numbers.

Real-World Applications

• Comparison symbols are used in various real-world scenarios.
• These include finance, science, engineering, and everyday problem-solving.

Common Mistakes

• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
• Incorrectly interpreting compound inequalities.
• Misunderstanding the properties of absolute value.