ALGEBRA

LINEAR EQUATIONS AND INEQUALITIES

Linear equations and inequalities are mathematical statements where variables have a highest power of

Linear Equations

A linear equation uses an equal sign (=) and is solved to find a specific value of the variable.

Example:

x + 3 = 7

x = 4

Solving Linear Equations

1. Isolate the variable by using inverse operations.

2. Whatever you do to one side, do the same to the other side.

3. Continue until the variable is by itself.

Example:

x + 5 = 12

Subtract 5 from both sides:

x = 7

Linear Inequalities

A linear inequality compares values using:

- < → less than

- > → greater than

- ≤ → less than or equal to

- ≥ → greater than or equal to

Instead of usually having one specific answer, an inequality can have a range of solutions.

Example:

x < 3

This means x can be any number less than 3.

Solving Linear Inequalities

- Use the same basic steps as solving equations.

- Keep the inequality balanced by doing the same operation to both sides.

Negative Rule

When multiplying or dividing both sides by a negative number, flip the direction of the inequality sign.

Example:

−2x > 6

Divide both sides by −2 and flip the sign:

x < −3

Equations vs. Inequalities

Linear equation → uses = → usually gives a specific solution.

Linear inequality → uses <, >, ≤, or ≥ → gives a range of possible solutions.

Inequalities can be represented on a number line or coordinate plane.

Quick Review

Equation → equal sign → specific value

Inequality → comparison sign → range of values

Isolate the variable → use inverse operations

Same operation on both sides → keeps it balanced

Multiply/divide by a negative → flip the inequality sign

FUNCTIONS

A function is a rule that assigns each input value to exactly one output value.

A function can be thought of as a machine: you put an input in, the function applies a rule, and you get an output.

Parts of a Function

Input

- The value put into the function.

- Usually represented by x.

Output

- The value produced by the function.

- Usually represented by f(x) or y.

Example:

f(x) = 2x + 1

If x = 3:

f(3) = 2(3) + 1 = 7

So:

- Input = 3

- Rule = 2x + 1

- Output = 7

Domain

- The set of all possible input values of a function.

Range

- The set of all possible output values of a function.

Rule

- The mathematical operation that changes the input into the output.

- Example: f(x) = 2x + 1

Function Rule

For something to be a function, each input must have only one output.

Example:

1 → 5

2 → 7

3 → 9

This is a function because every input has exactly one output.

If one input gives two different outputs, it is not a function.

Common Types of Functions

Linear Function

- Produces a straight-line graph.

- General form: f(x) = mx + b

Quadratic Function

- Produces a U-shaped graph called a parabola.

- Common form: f(x) = x²

Trigonometric Functions

- Produce repeating, wave-like patterns.

- Examples: sin(x) and cos(x)

Exponential Function

- Shows rapid growth or decay.

- General form: f(x) = aˣ

Quick Review

Function → input gives exactly one output

Input → value put into the function

Output → value produced

Domain → all possible inputs

Range → all possible outputs

Rule → operation that changes input into output

Linear → straight line

Quadratic → parabola

Trigonometric → repeating waves

Exponential → rapid growth or decay

Quadratic Equations

A quadratic equation is a second-degree polynomial equation with one variable. The highest power of the variable is 2.

Standard Form

ax² + bx + c = 0

Where:

- a = coefficient of x²; a ≠ 0

- b = coefficient of x

- c = constant term

Example:

2x² + 5x + 3 = 0

- a = 2

- b = 5

- c = 3

Quadratic Formula

The quadratic formula can be used to solve any quadratic equation in standard form.

x = (−b ± √(b² − 4ac)) / 2a

The ± means there can be two answers:

- one using +

- one using −

Methods for Solving Quadratic Equations

Factoring

- Rewrite the quadratic as a product of two factors.

- Set each factor equal to zero.

- Solve for the values of x.

Quadratic Formula

- Identify a, b, and c.

- Substitute them into the quadratic formula.

- Simplify to find the solutions.

Completing the Square

- Rearrange the equation to create a perfect square.

- Then solve for x.

Graphing

- Graph the corresponding quadratic function.

- The points where the parabola crosses the x-axis are the solutions.

Discriminant

The discriminant is the part inside the square root of the quadratic formula:

b² − 4ac

It tells you the number and type of solutions.

Positive Discriminant

b² − 4ac > 0

- Two distinct real solutions.

Zero Discriminant

b² − 4ac = 0

- One real solution that is repeated.

Negative Discriminant

b² − 4ac < 0

- Two complex solutions.

Quick Review

Quadratic → highest power is 2

Standard form → ax² + bx + c = 0

a → coefficient of x²

b → coefficient of x

c → constant

Quadratic formula → works for any quadratic equation

Discriminant → b² − 4ac

Positive → 2 real solutions

Zero → 1 repeated real solution

Negative → 2 complex solutions