Algebra: Equations and Inequalities

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What do you use when solving equations and inequalities?

Use inverse operations

Which of the following represents the inequality sign for greater than?

≥

> (correct)

≤

Which of the following represents the inequality sign for less than?

≤

< (correct)

≥

What is the inequality sign for greater than or equal to?

≥ Signup and view all the answers

What is the inequality sign for less than or equal to?

≤ Signup and view all the answers

What do you do when multiplying or dividing by a negative in inequalities?

Flip the inequality sign Signup and view all the answers

What does one solution to an equation look like?

X = # Signup and view all the answers

What does a no solution equation look like?

12 = 8 Signup and view all the answers

What does infinite solutions to equations look like?

13 = 13 Signup and view all the answers

What is the first step to solve the equation 3(2x+3)=6x+18+2x?

Clean up the equation (distribute & combine like terms) Signup and view all the answers

How do you solve the inequality -8x + 7 > 35?

Subtract 7, then divide by -8 and flip the inequality sign. Signup and view all the answers

What is the decay factor in the equation y = p(l - r)^t?

Number inside parentheses; 0 < df < 1 Signup and view all the answers

What is the name of a quadratic function?

Parabola Signup and view all the answers

What is the maximum or minimum point of a quadratic?

Vertex Signup and view all the answers

How do you find the vertex of a quadratic?

x = -b/2a; then substitute x into the equation Signup and view all the answers

What are the zeros or roots of a function?

The x-intercept; sometimes known as the solution Signup and view all the answers

How do you find the zeros or roots of a function?

<ol> <li>Square Root Method, 2. Factoring, 3. Quadratic formula, 4. Complete the square</li> </ol> Signup and view all the answers

What is the standard form of a quadratic?

y = ax² + bx + c Signup and view all the answers

What is vertex form?

y = a(x -/+ h)^2 + k Signup and view all the answers

What is factored form?

y = (x -/+ p)(x -/+ q) Signup and view all the answers

What is the axis of symmetry?

x value of the vertex; x = -b/2a Signup and view all the answers

What are the solutions of two or more functions?

Where they intersect Signup and view all the answers

How do you find the rate of change over a given interval?

Find where the two points are on the given graph & then find slope (rise/run) Signup and view all the answers

How do you transform a quadratic?

y = a(x -/+ h)² -/+ k Signup and view all the answers

What is the discriminant of a quadratic?

b² - 4ac Signup and view all the answers

Study Notes

Equations and Inequalities

Use inverse operations to solve equations and inequalities.
Inequality signs:
- Greater than: >
- Less than: <
- Greater than or equal to: ≥
- Less than or equal to: ≤
When multiplying or dividing by a negative in inequalities, flip the inequality sign.

Solutions to Equations

One solution example: (X = #)
No solution example: (12 = 8)
Infinite solutions example: (13 = 13)

Solving Equations

To solve (3(2x+3)=6x+18+2x), distribute, combine like terms, and isolate (x) to find (x = -4.5).
To solve (-8x+7>35), subtract, divide, and remember to flip the inequality when dividing by a negative.

Quadratic Functions

The decay factor in the formula (y = p(l - r)^t) must be between 0 and 1.
A quadratic function is represented as a parabola.
The vertex indicates the maximum or minimum point of a quadratic function.

Finding the Vertex

Compute vertex using (x = -\frac{b}{2a}); substitute (x) back into the equation for (y).

Zeros and Roots of Functions

The zeros or roots are the x-intercepts, where the function crosses the x-axis.
Methods to find zeros/roots:
- Square Root Method
- Factoring
- Quadratic formula
- Completing the square

Forms of Quadratic Functions

Standard form: (y = ax^2 + bx + c).
Vertex form: (y = a(x - h)^2 + k).
Factored form: (y = (x - p)(x - q)).
Axis of symmetry is found at (x = -\frac{b}{2a}).

Intersections of Functions

Solutions of two or more functions are found where they intersect.

Rate of Change

To find the rate of change over a given interval, identify the two points on the graph and calculate the slope ((rise/run)).

Transforming Quadratics

Transformation equation: (y = a(x \pm h)^2 \pm k).
- (+h): move left
- (-h): move right
- (+a): opens upward
- (-a): opens downward

Discriminant of a Quadratic

Calculated by (b^2 - 4ac):
- Positive: two solutions
- Negative: no real solutions
- Zero: one solution

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz covers key concepts related to solving equations and inequalities using inverse operations. It includes topics such as inequality signs, examples of solutions, and methods for solving linear and quadratic functions. Test your understanding of these fundamental algebraic principles.