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Questions and Answers
What operation is the inverse of multiplying by 4 when solving for a in the equation $4a = 3$?
What operation is the inverse of multiplying by 4 when solving for a in the equation $4a = 3$?
- Adding 4
- Subtracting 4
- Dividing by 4 (correct)
- Squaring
What is the inverse operation of squaring when solving for a in the equation $a^2 = 16$?
What is the inverse operation of squaring when solving for a in the equation $a^2 = 16$?
- Dividing by 2
- Square rooting (correct)
- Multiplying by 2
- Cubing
To keep an equation balanced, what action must be taken when an operation is performed on one side?
To keep an equation balanced, what action must be taken when an operation is performed on one side?
- Multiply the other side by two.
- Perform a different operation on the other side.
- Perform the same operation on the other side. (correct)
- Do nothing to the other side.
To solve for $x$ in the equation $x + 2.4 = 6.5$, what is the initial inverse operation?
To solve for $x$ in the equation $x + 2.4 = 6.5$, what is the initial inverse operation?
What is the first step in solving an algebraic equation with multiple operations acting on a variable?
What is the first step in solving an algebraic equation with multiple operations acting on a variable?
When solving a multi-step equation, in what order should you 'undo' addition and multiplication?
When solving a multi-step equation, in what order should you 'undo' addition and multiplication?
Why is it important to simplify an equation before reversing operations?
Why is it important to simplify an equation before reversing operations?
What is the main purpose of using inverse operations while solving equations?
What is the main purpose of using inverse operations while solving equations?
When using the balance strategy to solve an equation, what must be maintained?
When using the balance strategy to solve an equation, what must be maintained?
Under what condition is it not appropriate to use a balance scale model for solving equations?
Under what condition is it not appropriate to use a balance scale model for solving equations?
What is the primary goal when solving equations with variables on both sides?
What is the primary goal when solving equations with variables on both sides?
When solving equations with brackets, what is often the initial step?
When solving equations with brackets, what is often the initial step?
Why is it important to eliminate denominators when solving equations with fractions?
Why is it important to eliminate denominators when solving equations with fractions?
When eliminating denominators in an equation, what number should you multiply each term by?
When eliminating denominators in an equation, what number should you multiply each term by?
What does an inequality allow you to model that an equation does not?
What does an inequality allow you to model that an equation does not?
What is a key characteristic of the solution to an inequality?
What is a key characteristic of the solution to an inequality?
When graphing an inequality on a number line, what does an open circle indicate?
When graphing an inequality on a number line, what does an open circle indicate?
What does a closed circle signify when graphing an inequality on a number line?
What does a closed circle signify when graphing an inequality on a number line?
What is the correct way to represent "a number is greater than 5" as an inequality?
What is the correct way to represent "a number is greater than 5" as an inequality?
What is the correct way to represent "a number is less than or equal to 4" as an inequality?
What is the correct way to represent "a number is less than or equal to 4" as an inequality?
What type of data involves only whole numbers and is represented by solid dots on a number line?
What type of data involves only whole numbers and is represented by solid dots on a number line?
When continuous data is graphed on a number line, what visual representation is used?
When continuous data is graphed on a number line, what visual representation is used?
What happens to the inequality symbol when you multiply or divide both sides of an inequality by a negative number?
What happens to the inequality symbol when you multiply or divide both sides of an inequality by a negative number?
What is the first step when solving linear inequalities using addition or subtraction?
What is the first step when solving linear inequalities using addition or subtraction?
In the context of solving inequalities, what does verifying the solution entail?
In the context of solving inequalities, what does verifying the solution entail?
What does the phrase 'at least 10 people' translate to in inequality notation, where $x$ represents the number of people?
What does the phrase 'at least 10 people' translate to in inequality notation, where $x$ represents the number of people?
If $x$ represents the age to drive in most provinces and it is stated you have to be 'at least 16 years old', which inequality represents this?
If $x$ represents the age to drive in most provinces and it is stated you have to be 'at least 16 years old', which inequality represents this?
Which symbol represents 'is less than or equal to'?
Which symbol represents 'is less than or equal to'?
Flashcards
Inverse Operations
Inverse Operations
Operations that reverse or "undo" each other, bringing you back to the starting point.
Using Inverse Operators
Using Inverse Operators
A method to isolate a variable in an equation by performing the opposite operation.
Balanced Equation
Balanced Equation
The concept of maintaining equality by performing the same operation on both sides of an equation.
Eliminating Denominators
Eliminating Denominators
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Inequality
Inequality
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Solution of an Inequality
Solution of an Inequality
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Open Circle (Inequalities)
Open Circle (Inequalities)
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Graphing Inequalities
Graphing Inequalities
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Reversing Inequality Symbol
Reversing Inequality Symbol
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Study Notes
- Inverse operations "undo" or reverse each other's results.
- Inverse operations will take you back to the starting point.
- Addition cancels Subtraction, and Multiplication cancels Division.
- The goal in algebra is to isolate the variable using inverse operations.
Inverse Operators
- Undoing operations in algebra involves the use of inverse operators.
- The inverse of addition is subtraction.
- Example: If a + 3 = 2, then undo +3 by subtracting 3, giving a = 2 - 3, hence a = -1.
- The inverse of subtraction is addition.
- Example: If a - 5 = 6, then undo -5 by adding 5, giving a = 6 + 5, hence a = 11.
- The inverse of multiplication is division.
- Example: If 4a = 3, then undo multiplication by 4 by dividing by 4, giving a = 3/4.
- The inverse of division is multiplication.
- Example: If a/7 = 6, then undo division by 7 by multiplying by 7, giving a = 6 x 7, hence a = 42.
- The inverse of squaring is square rooting.
- Example: If a² = 16, then undo the squaring by square rooting, giving a = √16, hence a = 4.
- The inverse of square rooting is squaring
- Example: If √a = 5, then undo the square rooting by squaring, giving a = 5², i.e., a = 5 x 5, hence a = 25.
Solving equations tips
- Use inverse operations to solve equations.
- Determine operations applied to the variable to "build" the equation.
- Use the inverse operation to isolate the variable.
- Maintain balance by performing the same operation on both sides of the equation.
- Follow the order: addition/subtraction first, then multiplication/division.
- Verify solutions by substituting back into the equation.
Multi-step equations
- Apply the reverse order of operations to reverse operations in algebraic equations.
- Reverse addition and subtraction outside brackets first.
- Reverse multiplication and division outside brackets second.
- Remove outermost brackets and reverse operations within, following the correct order.
- Perform inverse operations in reverse order to "undo" a sequence of operations.
Solving Equations with Brackets
- Simplify the equation before reversing operations, combining like terms first.
- Two methods for solving equations with brackets include expanding the brackets first or reversing the process directly.
- Both methods yield the same answer, allowing a choice based on preference.
Isolating Values
- Focus on isolating the variable on one side and numbers on the other.
- "Remember that the goal is to get the variable on one side of the equation and the numbers on the other side of the equation!"
Equations using the Distributive Property
- Apply the distributive property to solve equations.
- Example of distribution: B(A + Y) = BA + BY
- Utilize the distributive property to simplify and solve equations.
Equations with Variable on Both Sides
- To solve an equation with unknown letters on both sides, add or subtract to get the unknown on one side of the equation only.
- To avoid getting negative x terms, always remove the smaller number of x from both sides. Example: Solve 4x + 3 = 2x + 9 => => 4x + 3 = 2x + 9 => 2x + 3 = 9 => 2x = 6 => x = 3
Balance Strategy
- An equation can be isolated using a balance strategy.
- The equation is "balanced" when whatever we do to one side of the "scale"/equation, we must do to the other side.
- Isolate the variable, add/subtract and multiply/divide evenly on both sides.
Using Tiles
- Algebra tiles are not efficient with large numbers or fractions/decimals
- Isolate the variables, add/subtract and multiply/divide evenly on both sides.
Equations with Fractions
- Eliminate denominators by multiplying each term with a common denominator.
- Remove all fractions to simplify the equation.
Introduction to Linear Inequalities
- Inequalities model situations described by a range of numbers, not a single number.
- Symbols used include: > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to).
- Inequalities can show the time, t, for which a car could park legally.
- Linear equations are true for only one value, but inequalities can be true for many.
Graphing Inequalities on a Number Line
- Open Circle: The value is not included.
- Filled Circle: The value is included.
- Less Than<open circle
- Greater Than>open circle
- Less Than or Equal to ≤closed circle
- Greater Than or Equal to ≥closed circle
Rules when Graphing Inequalities
-
or < use hollow dots on the number line.
- ≥ or ≤ use solid dots on the number line.
- For continuous data, shade the line.
- For discrete data, use dots on whole numbers.
Rules when working with Inequalities
- When multiplication or division by a negative number occurs, reverse the inequality symbol
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