Certainly! Below is a detailed breakdown of the key terms and concepts from Unit 2: Solving Equations and Inequalities, including definitions, examples with steps, and visuals where applicable.
U2 L1: Expression
Meaning: An expression is a combination of numbers, variables, and operation symbols (like +, -, *, /) without an equality sign.
Example:
- Expression: \( 3x + 5 \)
Steps:
- Identify the variable: \( x \)
- Identify the coefficient: 3
- Identify the constant: 5
U2 L1: Terms
Meaning: Parts of an expression separated by + or - signs.
Example:
- Expression: \( 4y - 2 + 7 \)
- Terms: \( 4y, -2, 7 \)
U2 L1: Factors
Meaning: Numbers or variables that are multiplied together.
Example:
- Expression: \( 6xy \)
- Factors: 6, \( x \), \( y \)
U2 L1: Coefficients
Meaning: A coefficient is a numerical factor in a term.
Example:
- In the term \( 7x^2 \), the coefficient is 7.
Rational Coefficient? A coefficient that can be expressed as a fraction \( p/q \) where \( p \) and \( q \) are integers and \( q \neq 0 \).
U2 L2: Inverse Operations
Meaning: Operations that undo each other. For example, addition and subtraction are inverse operations.
U2 L2: Equation with One Variable - Solve with One Step
Example:
- Equation: \( x + 4 = 10 \)
Steps:
- Subtract 4 from both sides: \( x + 4 - 4 = 10 - 4 \)
- Simplify: \( x = 6 \)
U2 L3: Equation with One Variable - Solve with Two Steps
Example:
- Equation: \( 2x + 3 = 11 \)
Steps:
- Subtract 3 from both sides: \( 2x + 3 - 3 = 11 - 3 \) yields \( 2x = 8 \)
- Divide both sides by 2: \( x = 4 \)
U2 L4: Like Terms
Meaning: Terms that have the same variable raised to the same power.
Example:
- Expression: \( 3x + 5x - 2 \)
- Like terms: \( 3x \) and \( 5x \)
Combine: \( 3x + 5x = 8x \) → Final expression: \( 8x - 2 \)
U2 L4: Distributive Property
Meaning: A property that states \( a(b + c) = ab + ac \).
Example:
- Expression: \( 3(x + 4) \)
Steps:
- Apply distributive property: \( 3x + 12 \)
U2 L5: Equation with One Variable on Both Sides
Example:
- Equation: \( 2x + 3 = x + 7 \)
Steps:
- Subtract \( x \) from both sides: \( 2x - x + 3 = 7 \)
- Simplify: \( x + 3 = 7 \)
- Subtract 3: \( x = 4 \)
U2 L6: Real World Scenario Setting Up and Solving Equations
Example:
- Scenario: Tim has twice as many apples as Jerry. If Jerry has 5 apples, how many does Tim have?
Equation: \( T = 2J \)
Steps:
- Substitute \( J = 5 \): \( T = 2(5) \)
- Solve: \( T = 10 \) (Tim has 10 apples.)
U2 L7: Solution to an Equation
- How do you know how many solutions you will have?
- One Solution: Linear equation with one variable (e.g., \( x + 2 = 5 \))
- No Solution: Parallel lines (e.g., \( x + 2 = x + 3 \))
- Infinite Solutions: Identical equations (e.g., \( 2(x + 1) = 2x + 2 \))
U2 L8: Inequality with One Variable
Example:
- Inequality: \( x + 3 > 7 \)
Steps:
- Subtract 3: \( x > 4 \)
U2 L8: Solutions of Inequality
Difference when solving an equation or an inequality?
- Equations have exact solutions; inequalities have a range of solutions.
Inequality symbols:
- \( >, <, \geq, \leq \)
Inequality solution with a number line:
- Graph \( x > 4 \) with an open circle at 4 and a line extending to the right.
U2 L9: Justify Solving Steps Using 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 \( a \cdot c = b \cdot c \) and \( \frac{a}{c} = \frac{b}{c} \) (where \( c \neq 0 \)).
- Distributive Property: \( a(b + c) = ab + ac \).
- Commutative Property: \( a + b = b + a \) and \( ab = ba \).
- Combine Like Terms/Substitution Property: Replace like terms with a single term.
U2 L10: Rearranging Formulas to Solve for One of the Variables
Example:
- Formula: \( A = lw \) (Area of a rectangle)
- Solve for \( w \):
Steps:
- Divide both sides by \( l \): \( w = \frac{A}{l} \)
This structure provides a comprehensive guide through the concepts of solving equations and inequalities, and inclusively covers definitions, examples, and procedural steps for enhanced understanding.