Algebra 1: Unit 1 Equations & Inequalities

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Questions and Answers

What is an algebraic expression?

A mathematical phrase that includes one or more variables.

What is an algebraic equation?

A statement of equality between two expressions.

What is a number sentence?

A statement of equality between two numerical expressions.

What are truth values?

True and false statements. Signup and view all the answers

What is a variable?

A symbol, usually a letter, that represents one or more numbers. Signup and view all the answers

What are like terms?

Terms with exactly the same variable factors in a variable expression or equation. Signup and view all the answers

What is the addition property of equality?

If $a = b$, then $a + c = b + c$. Signup and view all the answers

What is the subtraction property of equality?

If $a = b$, then $a - c = b - c$. Signup and view all the answers

What is the multiplication property of equality?

If $a = b$, then $a(c) = b(c)$. Signup and view all the answers

What is the division property of equality?

If $a = b$ and $c$ does not equal zero, then $\frac{a}{c} = \frac{b}{c}$. Signup and view all the answers

What is a coefficient?

The numerical factor when a term has a variable. Signup and view all the answers

What is a solution in the context of equations?

Any value or values that make an equation true. Signup and view all the answers

What does 'no solution' mean?

No value that we could substitute for a variable to make an equation true. Signup and view all the answers

What does 'infinite many solutions' refer to?

All real numbers; no matter what number we substitute for a variable, the equation would still be true. Signup and view all the answers

What does 'viable' mean in terms of equations?

The solution or no solution to an equation works successfully. Signup and view all the answers

What does 'non-viable' mean?

The solution or no solution to an equation does not work successfully. Signup and view all the answers

What are constraints?

A limitation or restriction that would make an equation not true. Signup and view all the answers

What is the Distributive Property?

A property of real numbers that states that the product of a number and the sum or difference of two numbers is the same as the sum or difference of their products. Signup and view all the answers

What is the Commutative Property?

The order in which numbers are added or multiplied does not affect their sum or product. Signup and view all the answers

What is a compound inequality?

The result of two simple inequalities that are combined by the words AND or OR. Signup and view all the answers

What is the Associative Property?

The sum or product of terms has the same value, regardless of how the terms are grouped. Signup and view all the answers

What is a rational number?

Any number that can be expressed as a fraction of two integers. Signup and view all the answers

What are irrational numbers?

Any number that cannot be expressed as a fraction; real numbers that are non-repeating, non-terminating decimals. Signup and view all the answers

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Study Notes

Algebraic Concepts

Algebraic Expression: A combination of numbers, variables, and operations. Example: 4x + 13.
Algebraic Equation: An equation where two algebraic expressions are set equal to each other. Example: 7x + 3 = 24.
Number Sentence: Similar to an algebraic equation but involves numerical expressions only. Example: 4x + 13 = 7x + 3.

Logical Foundations

Truth Values: Indicates whether a statement is true or false; example: 7x + 3 = 24 is true when x = 3.

Variables and Terms

Variable: A symbol (typically a letter) representing one or more numbers. Common variables include x, y, z.
Like Terms: Terms that share the same variable factors; for example, 4x and -7x can be combined.

Properties of Equality

Addition Property of Equality: Adding the same number to both sides of an equation keeps it balanced (If a = b, then a + c = b + c).
Subtraction Property of Equality: Subtracting the same number from both sides maintains equality (If a = b, then a - c = b - c).
Multiplication Property of Equality: Multiplying both sides of an equation by the same number keeps them equal (If a = b, then a(c) = b(c)).
Division Property of Equality: Dividing both sides by the same non-zero number preserves equality (If a = b and c ≠ 0, then a/c = b/c).

Solutions in Equations

Coefficient: The numerical factor in a term with a variable. For instance, in 4x, 4 is the coefficient.
Solution: A value or values that satisfy the equation. Example: x = 3 is a solution for 7x + 3 = 24.
No Solution: There is no value that can satisfy the equation, for example, 5x + 6 = 5x - 3 yields no solution.
Infinite Solutions: The equation holds true for all real numbers. Example: 5x + 6 = 5x + 6.

Viability of Solutions

Viable: Solutions or no solutions that work successfully in an equation.
Non-Viable: Solutions or no solutions that do not satisfy the equation.

Constraints

Constraints: Limitations or restrictions that prevent an equation from being true.

Properties of Operations

Distributive Property: States that a(b + c) = ab + ac; relevant for simplifying expressions involving multiplication over addition or subtraction.
Commutative Property: The order of addition or multiplication does not affect the result; a + b = b + a.
Associative Property: Grouping of terms does not affect the sum or product; (a + b) + c = a + (b + c).

Types of Numbers

Rational Numbers: Any number expressible as a fraction of two integers; for instance, 1/2 or -3/4.
Irrational Numbers: Numbers that cannot be expressed as fractions; examples include non-repeating, non-terminating decimals.