Questions and Answers
Which property states that the order in which two numbers are added does not affect the sum?
According to the associative law of addition, what is the correct transformation of the expression 3 + (4 + 5)?
What does the distributive law state when applying a number to a sum?
Which inequality symbol best represents the statement where a is less than or equal to b?
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What is an outcome of the transitive property of equality?
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What describes the values of x in the compound inequality x > 1 and x ≤ 5?
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What is the result when both sides of an inequality are multiplied by a negative number?
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What theorem states that adding the same number to both sides of an inequality does not change the inequality's direction?
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If a is greater than b and both sides are multiplied by 3, which inequality is true?
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Which of the following statements about the compound inequality x < 1 or x > 5 is true?
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Study Notes
Basic Laws of Natural Numbers
- Closure Law for Addition: For any numbers a and b, their sum a + b results in another number z (e.g., 2 + 5 = 7).
- Commutative Law for Addition: The order of numbers does not affect the sum (e.g., 5 + 7 = 7 + 5 = 12).
- Associative Law for Addition: Grouping of addends does not affect the sum (e.g., 12 + (13 + 22) = (12 + 13) + 22).
- Closure Law for Multiplication: The product of any two numbers a and b is also a number (e.g., 2 x 4 = 8).
- Commutative Law for Multiplication: The order of factors does not affect the product (e.g., 12 x -3 = -3 x 12).
- Associative Law for Multiplication: Grouping of factors does not affect the product (e.g., 6 (-4 x -3) = (6 x -4) x -3).
- Distributive Law: A number multiplied by the sum or difference of two numbers can be distributed (e.g., 3(5 + 8) = 39, 3(5 - 8) = -9).
Basic Laws of Equality
- Reflexive Property: Any number is equal to itself (a = a).
- Symmetric Property: If one number equals another, the reverse is also true (if a = b, then b = a).
- Transitive Property: If a equals b and b equals c, then a equals c.
- Addition of Equals: If a = b and c = d, then a + c = b + d.
- Multiplication of Equals: If a = b and c = d, then ac = bd.
Inequality
- Definition: An inequality compares two quantities indicating one is greater than or less than the other (e.g., a > b).
- Compound Inequality: Represents two inequalities connected by "or" or "and" (e.g., 1 < x ≤ 5).
Theorems of Inequalities
- Theorem 1: Adding the same number to both sides of an inequality maintains the inequality (e.g., if a > b, then a + c > b + c).
- Theorem 2: Multiplying both sides by a positive number preserves the inequality (e.g., if a > b and c > 0, then ac > bc).
- Theorem 3: Multiplying both sides by a negative number reverses the inequality (e.g., if a > b and c < 0, then ac < bc).
- Theorem 4: Changing the signs on both sides of an inequality reverses the inequality (e.g., if a > b, then -a < -b).
- Theorem 5: Taking reciprocals of both sides of a positive or negative inequality also reverses the sense of inequality (e.g., if a > b, then 1/a < 1/b).
Solving Inequalities
- Standard Form: A linear inequality is expressed as ax + b < c. If a is positive, the method to solve is similar to solving equations (e.g., x < (c - b) / a).
- Example Solutions: For the inequality 4x + 7 < 31, solve for x yields x < 6. For -2x + 7 < 23, the solution leads to x > -8.
Compound Inequalities
- Represent two interconnected inequalities. These require understanding how to solve each segment while considering their relationship.
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Description
Explore the fundamental laws governing natural numbers and equality in this quiz. You'll learn about closure, commutative, associative, and distributive laws along with properties of equality. Perfect for anyone looking to strengthen their understanding of basic mathematical principles.
