Question
Unit 2 Solving Equations and Inequalities
Vocabulary
Key Terms
Meaning - Definition - Picture
Example with problems solved showing all steps
U2 L1 Expression
U2 L1
Terms
U2 L1
Factors
U2 L1
Coefficients
Rational Coefficient?
U2 L2
Inverse Operations
U2 L2 Equation with one variable
Solve with one step
U2 L3
Equation with one variable
Solve with two steps
U2 L4
Like Terms
U2 L4
Distributive Property
U2 L5
Equation with one variable on both sides
U2 L6
Real world scenario setting up and solving equations
U2 L7
Solution to an equation
How do you know how many solutions you will have?
One solution
No solution
Infinite solutions
U2 L8
Inequality with one variable
U2 L8
Solutions of Inequality
Difference when solving an equation or an inequality?
Inequality solutions using symbols
Inequality solution with a numberline
U2 L8
Inequality Extension Notes
Inequality Extension Notes Link - Make your copy and complete
U2 L9
Justify solving steps using properties
Addition and Subtraction Property of Equality
Multiplication and Division Property of Equality
Distributive Property
Commutative Property
Combine Like terms- Substitution Property
U2 L10
Rearranging formulas to solve for one of the variables
Vocabulary
Key Terms
Meaning - Definition - Picture
Example with problems solved showing all steps
U2 L1 Expression
U2 L1
Terms
U2 L1
Factors
U2 L1
Coefficients
Rational Coefficient?
U2 L2
Inverse Operations
U2 L2 Equation with one variable
Solve with one step
U2 L3
Equation with one variable
Solve with two steps
U2 L4
Like Terms
U2 L4
Distributive Property
U2 L5
Equation with one variable on both sides
U2 L6
Real world scenario setting up and solving equations
U2 L7
Solution to an equation
How do you know how many solutions you will have?
One solution
No solution
Infinite solutions
U2 L8
Inequality with one variable
U2 L8
Solutions of Inequality
Difference when solving an equation or an inequality?
Inequality solutions using symbols
Inequality solution with a numberline
U2 L8
Inequality Extension Notes
Inequality Extension Notes Link - Make your copy and complete
U2 L9
Justify solving steps using properties
Addition and Subtraction Property of Equality
Multiplication and Division Property of Equality
Distributive Property
Commutative Property
Combine Like terms- Substitution Property
U2 L10
Rearranging formulas to solve for one of the variables
Answers
GPT-4o mini
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:**
1. Identify the variable: \( x \)
2. Identify the coefficient: 3
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:**
1. Subtract 4 from both sides: \( x + 4 - 4 = 10 - 4 \)
2. Simplify: \( x = 6 \)
---
### **U2 L3: Equation with One Variable - Solve with Two Steps**
**Example:**
- Equation: \( 2x + 3 = 11 \)
**Steps:**
1. Subtract 3 from both sides: \( 2x + 3 - 3 = 11 - 3 \) yields \( 2x = 8 \)
2. 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:**
1. Apply distributive property: \( 3x + 12 \)
---
### **U2 L5: Equation with One Variable on Both Sides**
**Example:**
- Equation: \( 2x + 3 = x + 7 \)
**Steps:**
1. Subtract \( x \) from both sides: \( 2x - x + 3 = 7 \)
2. Simplify: \( x + 3 = 7 \)
3. 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:**
1. Substitute \( J = 5 \): \( T = 2(5) \)
2. 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:**
1. 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:**
1. 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.
---
### **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:**
1. Identify the variable: \( x \)
2. Identify the coefficient: 3
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:**
1. Subtract 4 from both sides: \( x + 4 - 4 = 10 - 4 \)
2. Simplify: \( x = 6 \)
---
### **U2 L3: Equation with One Variable - Solve with Two Steps**
**Example:**
- Equation: \( 2x + 3 = 11 \)
**Steps:**
1. Subtract 3 from both sides: \( 2x + 3 - 3 = 11 - 3 \) yields \( 2x = 8 \)
2. 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:**
1. Apply distributive property: \( 3x + 12 \)
---
### **U2 L5: Equation with One Variable on Both Sides**
**Example:**
- Equation: \( 2x + 3 = x + 7 \)
**Steps:**
1. Subtract \( x \) from both sides: \( 2x - x + 3 = 7 \)
2. Simplify: \( x + 3 = 7 \)
3. 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:**
1. Substitute \( J = 5 \): \( T = 2(5) \)
2. 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:**
1. 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:**
1. 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.