Questions and Answers
What property is used to prove that (-a) b = a (-b) = -ab?
What is the result of multiplying a by (-b)?
If a and b are real numbers, what can be concluded about the product (-a) (-b)?
Which of the following is an example of the closure property of multiplication?
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If a and b are real numbers, what can be concluded about the expression a (-b)?
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Which of the following is an example of the associative property of multiplication?
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What is the result of multiplying (-a) by b?
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If a, b, c, and d are real numbers, what can be concluded about the expression c (bd)?
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Which of the following illustrates the closure property of multiplication?
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What is the result of (4 % 5) % 8?
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Which property is used in the equation -3 < -2 ⇒ 0 < 1?
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Which of the following is a consequence of the multiplicative properties of inequality?
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What is the distributive property of multiplication over addition?
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Which of the following is an example of a set operation?
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What is the value of x in the equation x - x = 0?
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What is the result of 1000 % 1?
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Which of the following is a property of a field?
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What is the additive inverse of a number a?
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Which of the following sets possesses all the properties of a field?
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What is the result of the set operation a ∪ (b ∩ c)?
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What is the result of multiplying (a + b)c?
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Which of the following is a consequence of the order properties of inequality?
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Study Notes
Quadratic Equations and Number Systems
Properties of Real Numbers
- Addition and multiplication of real numbers are commutative, associative, and distributive.
- Properties of inequality:
- If both sides of an inequality are multiplied by a positive number, the direction of the inequality does not change.
- If both sides of an inequality are multiplied by a negative number, the direction of the inequality is reversed.
Properties of Multiplication
- Distributivity of multiplication over addition:
a(b + c) = ab + ac
and(a + b)c = ac + bc
- Properties of equality of numbers:
-
a
and(-a)
are additive inverses of each other. -
a(b - c) = ab - ac
and(x - y)z = xz - yz
Order Properties
- If
a > b
andc > d
, thenac > bd
. - If
a < b
andc < d
, thenac < bd
.
Properties of Inequality
- If both sides of an inequality are multiplied by a positive number, the direction of the inequality does not change.
- If both sides of an inequality are multiplied by a negative number, the direction of the inequality is reversed.
Examples and Applications
-
(-a)b = a(-b) = -ab
-
(-a)(-b) = ab
-
a(b + c - d) = ab + ac - ad
-
1000 % 1 = 1000
-
4.1 + (-4.1) = 0
-
a - a = 0
-
a(b - c) = ab - ac
and(x - y)z = xz - yz
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Test your understanding of algebra concepts, including quadratic equations, number systems, and multiplication addition law.