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 \( x + 5 = 12 \)
- Subtract 5 from both sides:
\( x + 5 - 5 = 12 - 5 \)
\( x = 7 \)
U2 L3: Equation with One Variable (Two Steps)
Example: Solve \( 2x + 3 = 11 \)
- Subtract 3 from both sides:
\( 2x + 3 - 3 = 11 - 3 \)
\( 2x = 8 \) - 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 \( 3x + 5 = 2x + 10 \)
- Subtract \( 2x \):
\( 3x - 2x + 5 = 10 \)
\( x + 5 = 10 \) - 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 \( x + 3 < 10 \)
- 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 \( l \).
- 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!
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