What is the primary difference between equations and inequalities in mathematics?
Equations state equality
Equations state that two expressions are equal
Inequalities provide a range of values as solutions.
Which symbols are used in inequalities to show unequal relationships?
<, >, ≤, ≥, ≠
Equations use the equals sign to show that both sides have the same value.
What type of solutions do equations typically yield?
Specific solutions
Steps to solve linear equations
1️⃣ Isolate the variable
2️⃣ Perform inverse operations
3️⃣ Simplify the solution
Quadratic equations are of the form ax^2 + bx + c = 0
Match the method to its description for solving quadratic equations:
Factoring ↔️ Find two numbers that multiply to ac and add to b
Completing the Square ↔️ Rearrange the equation to x^2 + bx = -c
Quadratic Formula ↔️ Use the formula x = (-b ± √(b^2 - 4ac))/(2a)
What is the quadratic formula used to solve quadratic equations?
x=2a−b±b2−4ac
The quadratic formula works for all quadratic equations.
What type of relationships do equations and inequalities describe?
Relationships between quantities
Match the feature with its corresponding category:
Definition ↔️ States that two expressions are equal for equations ||| Indicates that two expressions are not equal for inequalities
Solution Type ↔️ Specific value(s) for equations ||| Range of values for inequalities
Symbols Used ↔️ = for equations ||| <, >, ≤, ≥, ≠ for inequalities
Steps to solve linear equations
1️⃣ Isolate the variable
2️⃣ Perform inverse operations
3️⃣ Simplify the solution
What is the key difference between equations and inequalities in terms of solutions?
Specific values vs. ranges
Equations use the symbol '=', while inequalities use symbols like '<', '>', '≤', '≥', or '≠'.
Match the feature with its type:
Definition ↔️ States equality or inequality
Symbols Used ↔️ =, <, >, ≤, ≥, ≠
Solution Type ↔️ Specific value(s) or range
Equations state that two expressions are equal, while inequalities show unequal relationships.
Steps to solve linear equations
1️⃣ Isolate the variable
2️⃣ Perform inverse operations
3️⃣ Simplify the solution
What is the first step in solving the equation \( 2x - 3 = 5 \)?
Add 3 to both sides
After adding 3 to both sides and dividing by 2, the solution to \( 2x - 3 = 5 \) is \( x = 4
Solving linear equations involves isolating the variable on one side using inverse operations.
What operation is performed first to solve \( 2x - 3 = 5 \)?
Add 3 to both sides
The solution to the linear equation \( 2x - 3 = 5 \) is \( x = 4
What is the first step in solving the fractional linear equation \( \frac{1}{2}x - 1 = 3 \)?
Multiply by2
Quadratic equations are of the form \( ax^2 + bx + c = 0 \), where \( a \neq 0 \).
The factoring method for solving quadratic equations involves finding two numbers that multiply to \( ac \) and add to b.
Steps to solve a quadratic equation by factoring
1️⃣ Find two numbers that multiply to \( ac \) and add to \( b \)
2️⃣ Rewrite the equation in factored form
3️⃣ Solve for \( x \) to find the roots
What are the factors of 6 that add up to -5?
-2 and -3
Completing the square involves adding \( \left(\frac{b}{2}\right)^2 \) to both sides of the equation.
After adding \( \left(\frac{b}{2}\right)^2 \) to both sides, the left side of the equation can be factored into \( \left(x + \frac{b}{2}\right)^2 \), which is a perfect square.square.
What is the first step in completing the square for \( x^2 + 6x - 10 = 0 \)?
Rewrite as \( x^2 + 6x = 10 \)
The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
The quadratic formula requires identifying the values of \( a \), \( b \), and \( c \) from the quadratic equation.
What are the roots of the quadratic equation \( x^2 - 5x +6 = 0 \)?
2 and 3
Completing the square for \( x^2 + 6x - 10 = 0 \) results in \( (x + 3)^2 = 19 \).
The solutions for \( x \) in \( (x + 3)^2 = 19 \) are \( x = -3 \pm \sqrt{19} \), which involves taking the square root of both sides.
Rewrite the quadratic equation in factored form: (x - p)(x - q) = 0
What are the roots of the equation \( (x - p)(x - q) = 0 \)?