Here's a structured approach to your vocabulary for Unit 2 on Solving Equations and Inequalities, which includes definitions, example problems solved with all steps, and key concepts:
Unit 2: Solving Equations and Inequalities Vocabulary
U2 L1: Expression
Key Term Definition:
An expression is a mathematical phrase that includes numbers, variables (letters representing numbers), and operations (addition, subtraction, multiplication, division). Expressions do not have an equality sign and cannot be solved for a value, but they represent a quantity.
Example:
Expression: \( 3x + 5y - 2 \)
U2 L1: Terms
Definition:
Terms are the parts of an expression separated by addition or subtraction signs. Each term can be a number, a variable, or a product of numbers and variables.
Example:
In the expression \( 3x + 5y - 2 \):
- Terms: \( 3x \), \( 5y \), and \( -2 \)
U2 L1: Factors
Definition:
Factors are the numbers or expressions that are multiplied together to get a term or an expression.
Example:
In \( 3xy \), the factors are \( 3 \), \( x \), and \( y \).
U2 L1: Coefficients
Definition:
Coefficients are the numerical factors that multiply the variables in terms.
Rational Coefficient?:
A rational coefficient is a coefficient that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b \neq 0 \).
Example:
In \( 3x^2 + 4y \), the coefficients are \( 3 \) and \( 4 \).
U2 L2: Inverse Operations
Definition:
Inverse operations are operations that reverse the effect of each other. The main pairs are:
- Addition and subtraction
- Multiplication and division
Example:
To solve \( x + 5 = 12 \), use subtraction (inverse of addition):
- \( x + 5 - 5 = 12 - 5 \)
- \( x = 7 \)
U2 L2: Equation with One Variable (Solve with One Step)
Example Problem: Solve \( x + 7 = 10 \).
Solution Steps:
- Subtract 7 from both sides. \[ x + 7 - 7 = 10 - 7 \] \[ x = 3 \]
U2 L3: Equation with One Variable (Solve with Two Steps)
Example Problem: Solve \( 2x + 3 = 11 \).
Solution Steps:
- Subtract 3 from both sides. \[ 2x + 3 - 3 = 11 - 3 \] \[ 2x = 8 \]
- Divide both sides by 2. \[ x = 4 \]
U2 L4: Like Terms
Definition:
Like terms are terms in an expression that have the same variable raised to the same power.
Example:
In \( 5x + 2x - 3 \), the like terms are \( 5x \) and \( 2x \).
- Combined: \( 5x + 2x = 7x \)
- Final expression: \( 7x - 3 \)
U2 L4: Distributive Property
Definition:
The distributive property states that \( a(b + c) = ab + ac \).
Example Problem: Simplify \( 3(2x + 4) \).
Solution Steps:
- Apply the distributive property: \[ 3(2x) + 3(4) = 6x + 12 \]
U2 L5: Equation with One Variable on Both Sides
Example Problem:
Solve \( 3x + 4 = 2x + 10 \).
Solution Steps:
- Subtract \( 2x \) from both sides: \[ 3x - 2x + 4 = 10 \] \[ x + 4 = 10 \]
- Subtract 4 from both sides: \[ x = 6 \]
U2 L6: Real World Scenario Setting Up and Solving Equations
Example Scenario:
A book costs $10 more than a pen. If the pen costs \( p \), write an equation and solve if the pen costs $5.
Equation:
\( \text{Book} = p + 10 \)
If \( p = 5 \):
\[
\text{Book} = 5 + 10 = 15
\]
U2 L7: Solution to an Equation
Definition:
A solution is a value that makes an equation true.
How many solutions can you have?
- One solution
- No solution
- Infinite solutions
U2 L8: Inequality with One Variable
Example Problem:
Solve \( x + 3 < 7 \).
Solution Steps:
- Subtract 3 from both sides: \[ x < 7 - 3 \] \[ x < 4 \]
U2 L8: Solutions of Inequality
Difference When Solving an Equation or an Inequality:
Solving inequalities involves additional rules, especially when multiplying/dividing by negative numbers (flipping the inequality sign).
Inequality Solutions Using Symbols:
- \( < \) (less than), \( > \) (greater than)
- \( \leq \) (less than or equal to), \( \geq \) (greater than or equal to)
Inequality Solution with a Number Line:
To illustrate \( x < 4 \), shade to the left of 4 and use an open circle to show 4 is not included.
U2 L9: Justifying Solving Steps Using Properties
Properties:
-
Addition and Subtraction Property of Equality:
If \( a = b \), then \( a + c = b + c \) and \( a - c = b - c \). -
Multiplication and Division Property of Equality:
If \( a = b \), then \( ac = bc \) and \( \frac{a}{c} = \frac{b}{c} \) (if \( c \neq 0 \)). -
Distributive Property:
\( a(b + c) = ab + ac \). -
Commutative Property:
\( a + b = b + a \) and \( ab = ba \). -
Combine Like Terms - Substitution Property:
If \( a = b \), then \( a \) can be substituted for \( b \) in any expression.
U2 L10: Rearranging Formulas to Solve for One of the Variables
Example Formula:
Solve \( A = lw \) for \( l \).
Solution Steps:
- Divide both sides by \( w \): \[ A = lw \implies l = \frac{A}{w} \]
This structured approach should help in understanding the key concepts and terms related to solving equations and inequalities. Adjust the examples based on your learning needs.