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
Sure! Below you’ll find a detailed breakdown of Unit 2 on Solving Equations and Inequalities, including key terms, definitions, example problems worked out step-by-step, and other relevant information:
### U2 L1: Expression
**Meaning**: An expression is a combination of numbers, variables, and operators (like + and -) that represents a value.
**Example**:
- Expression: \( 3x + 5 \)
- Here, \( 3 \) is a coefficient of \( x \), and \( 5 \) is a constant.
### U2 L1: Terms
**Meaning**: Terms are the individual components of an expression. They can be constants, variables, or a combination of both.
**Example**: In \( 4x + 3y - 7 \),
- The terms are \( 4x \), \( 3y \), and \(-7\).
### U2 L1: Factors
**Meaning**: Factors are numbers or expressions that are multiplied together.
**Example**: In \( 6x^2 \), the factors are \( 6 \) and \( x^2 \).
### U2 L1: Coefficients
**Meaning**: Coefficients are numerical factors in front of variables.
**Example**: In \( 5x \), \( 5 \) is the coefficient of \( x \).
**Rational Coefficient**: A rational coefficient is a coefficient that can be expressed as a fraction or ratio of two integers.
### U2 L2: Inverse Operations
**Meaning**: Inverse operations are operations that undo each other (e.g., addition and subtraction, multiplication and division).
### U2 L2: Equation with One Variable (One Step)
**Example**: Solve \( x + 5 = 12 \)
1. Subtract 5 from both sides:
\( x + 5 - 5 = 12 - 5 \)
\( x = 7 \)
### U2 L3: Equation with One Variable (Two Steps)
**Example**: Solve \( 2x + 3 = 11 \)
1. Subtract 3 from both sides:
\( 2x + 3 - 3 = 11 - 3 \)
\( 2x = 8 \)
2. Divide both sides by 2:
\( \frac{2x}{2} = \frac{8}{2} \)
\( x = 4 \)
### U2 L4: Like Terms
**Meaning**: Like terms are terms that have the same variable raised to the same power.
**Example**: In \( 3x + 4x - 7 \), \( 3x \) and \( 4x \) are like terms.
### U2 L4: Distributive Property
**Meaning**: The distributive property states that \( a(b + c) = ab + ac \).
**Example**:
\( 3(x + 4) \) expands to
\( 3x + 12 \).
### U2 L5: Equation with One Variable on Both Sides
**Example**: Solve \( 3x + 5 = 2x + 10 \)
1. Subtract \( 2x \):
\( 3x - 2x + 5 = 10 \)
\( x + 5 = 10 \)
2. Subtract 5:
\( x = 5 \)
### U2 L6: Real World Scenario Setting Up and Solving Equations
**Example**: If a book costs $7 and you have $20, how many books can you buy?
- Let \( x \) be the number of books.
- Set up the equation:
\( 7x = 20 \)
Solve:
\( x = \frac{20}{7} \approx 2.86 \)
You can buy 2 books.
### U2 L7: Solution to an Equation
**Meaning**: A solution is a value that makes the equation true.
**How do you know how many solutions you will have?**:
- **One solution**: If it's a linear equation.
- **No solution**: If the lines are parallel (e.g., \( x + 1 = x + 2 \)).
- **Infinite solutions**: If the two sides are identical (e.g., \( 2x + 4 = 2(x + 2) \)).
### U2 L8: Inequality with One Variable
**Example**: Solve \( x + 3 < 10 \)
1. Subtract 3 from both sides:
\( x < 7 \)
### U2 L8: Solutions of Inequality
**Difference when solving an equation or an inequality**: When you multiply or divide an inequality by a negative number, you must reverse the inequality sign.
**Inequality Solutions Using Symbols**:
- \( < \) means 'less than'
- \( > \) means 'greater than'
- \( \leq \) means 'less than or equal to'
- \( \geq \) means 'greater than or equal to'
**Inequality Solution with a Number Line**: Show the solution on a number line indicating open and closed circles for inequalities.
### U2 L9: Justifying Solving Steps Using Properties
- **Addition and Subtraction Property of Equality**: If \( a = b \), then \( a + c = b + c \).
- **Multiplication and Division Property of Equality**: If \( a = b \), then \( ac = bc \).
- **Distributive Property**: \( a(b + c) = ab + ac \).
- **Commutative Property**: \( a + b = b + a \).
- **Combine Like Terms - Substitution Property**: Use known values to replace variables in expressions.
### U2 L10: Rearranging Formulas to Solve for One of the Variables
**Example**: Rearranging \( A = lw \) to solve for \( l \).
1. Divide by \( w \):
\( l = \frac{A}{w} \).
This format should help with understanding key concepts in Unit 2 about solving equations and inequalities. Feel free to ask if you have any questions or need additional explanations!
### U2 L1: Expression
**Meaning**: An expression is a combination of numbers, variables, and operators (like + and -) that represents a value.
**Example**:
- Expression: \( 3x + 5 \)
- Here, \( 3 \) is a coefficient of \( x \), and \( 5 \) is a constant.
### U2 L1: Terms
**Meaning**: Terms are the individual components of an expression. They can be constants, variables, or a combination of both.
**Example**: In \( 4x + 3y - 7 \),
- The terms are \( 4x \), \( 3y \), and \(-7\).
### U2 L1: Factors
**Meaning**: Factors are numbers or expressions that are multiplied together.
**Example**: In \( 6x^2 \), the factors are \( 6 \) and \( x^2 \).
### U2 L1: Coefficients
**Meaning**: Coefficients are numerical factors in front of variables.
**Example**: In \( 5x \), \( 5 \) is the coefficient of \( x \).
**Rational Coefficient**: A rational coefficient is a coefficient that can be expressed as a fraction or ratio of two integers.
### U2 L2: Inverse Operations
**Meaning**: Inverse operations are operations that undo each other (e.g., addition and subtraction, multiplication and division).
### U2 L2: Equation with One Variable (One Step)
**Example**: Solve \( x + 5 = 12 \)
1. Subtract 5 from both sides:
\( x + 5 - 5 = 12 - 5 \)
\( x = 7 \)
### U2 L3: Equation with One Variable (Two Steps)
**Example**: Solve \( 2x + 3 = 11 \)
1. Subtract 3 from both sides:
\( 2x + 3 - 3 = 11 - 3 \)
\( 2x = 8 \)
2. Divide both sides by 2:
\( \frac{2x}{2} = \frac{8}{2} \)
\( x = 4 \)
### U2 L4: Like Terms
**Meaning**: Like terms are terms that have the same variable raised to the same power.
**Example**: In \( 3x + 4x - 7 \), \( 3x \) and \( 4x \) are like terms.
### U2 L4: Distributive Property
**Meaning**: The distributive property states that \( a(b + c) = ab + ac \).
**Example**:
\( 3(x + 4) \) expands to
\( 3x + 12 \).
### U2 L5: Equation with One Variable on Both Sides
**Example**: Solve \( 3x + 5 = 2x + 10 \)
1. Subtract \( 2x \):
\( 3x - 2x + 5 = 10 \)
\( x + 5 = 10 \)
2. Subtract 5:
\( x = 5 \)
### U2 L6: Real World Scenario Setting Up and Solving Equations
**Example**: If a book costs $7 and you have $20, how many books can you buy?
- Let \( x \) be the number of books.
- Set up the equation:
\( 7x = 20 \)
Solve:
\( x = \frac{20}{7} \approx 2.86 \)
You can buy 2 books.
### U2 L7: Solution to an Equation
**Meaning**: A solution is a value that makes the equation true.
**How do you know how many solutions you will have?**:
- **One solution**: If it's a linear equation.
- **No solution**: If the lines are parallel (e.g., \( x + 1 = x + 2 \)).
- **Infinite solutions**: If the two sides are identical (e.g., \( 2x + 4 = 2(x + 2) \)).
### U2 L8: Inequality with One Variable
**Example**: Solve \( x + 3 < 10 \)
1. Subtract 3 from both sides:
\( x < 7 \)
### U2 L8: Solutions of Inequality
**Difference when solving an equation or an inequality**: When you multiply or divide an inequality by a negative number, you must reverse the inequality sign.
**Inequality Solutions Using Symbols**:
- \( < \) means 'less than'
- \( > \) means 'greater than'
- \( \leq \) means 'less than or equal to'
- \( \geq \) means 'greater than or equal to'
**Inequality Solution with a Number Line**: Show the solution on a number line indicating open and closed circles for inequalities.
### U2 L9: Justifying Solving Steps Using Properties
- **Addition and Subtraction Property of Equality**: If \( a = b \), then \( a + c = b + c \).
- **Multiplication and Division Property of Equality**: If \( a = b \), then \( ac = bc \).
- **Distributive Property**: \( a(b + c) = ab + ac \).
- **Commutative Property**: \( a + b = b + a \).
- **Combine Like Terms - Substitution Property**: Use known values to replace variables in expressions.
### U2 L10: Rearranging Formulas to Solve for One of the Variables
**Example**: Rearranging \( A = lw \) to solve for \( l \).
1. Divide by \( w \):
\( l = \frac{A}{w} \).
This format should help with understanding key concepts in Unit 2 about solving equations and inequalities. Feel free to ask if you have any questions or need additional explanations!