Precalculus Study Notes: Equations and Inequalities (Ch. 1)

Equations and Inequalities

Solving Linear Equations

Linear equations are foundational in algebra and precalculus, representing relationships with constant rates of change. A linear equation in one variable has the general form $ax + b = c$.

• Linear Expression: An algebraic expression of the form $ax + b$.

• Linear Equation: An equation set equal to a value, e.g., $2x + 3 = 7$.

• Solving for the Unknown: Find the value(s) of $x$ that make the equation true.

• Operations: Use addition, subtraction, multiplication, or division to isolate the variable.

Example: Solve $2x + 3 = 7$.

• Subtract 3 from both sides: $2x = 4$

• Divide both sides by 2: $x = 2$

Key Steps for Solving Linear Equations:

1. Eliminate parentheses

2. Combine like terms

3. Isolate the variable

4. Check solution in original equation

Linear Equations with Fractions

Some linear equations contain fractions. To solve, clear fractions by multiplying both sides by the Least Common Denominator (LCD).

• Multiply by LCD: Eliminates fractions.

• Combine like terms and solve: Proceed as with regular linear equations.

Example: Solve $\frac{x+2}{3} = \frac{1}{2}$.

• Multiply both sides by 6 (LCD): $2(x+2) = 3$

• Solve: $2x + 4 = 3 \implies 2x = -1 \implies x = -\frac{1}{2}$

Categorizing Linear Equations

Linear equations can be classified based on the number of solutions:

• One Solution (Conditional): The equation is true for one value of the variable.

• No Solution (Inconsistent): The equation is never true.

• Infinite Solutions (Identity): The equation is true for all values of the variable.

Type	Example	Solution
Conditional	$2x + 4 = 10$	$x = 3$
Identity	$2x + 5 = 2x + 5$	All real numbers
Inconsistent	$2x + 5 = 2x + 3$	No solution

Solving Rational Equations

A rational equation contains one or more rational expressions (fractions with polynomials in numerator and denominator). To solve:

• Multiply both sides by the LCD to eliminate denominators.

• Solve the resulting equation.

• Check for extraneous solutions (values that make any denominator zero).

Example: Solve $\frac{1}{x} - 1 = 12$.

• Add 1 to both sides: $\frac{1}{x} = 13$

• Invert: $x = \frac{1}{13}$

The Imaginary Unit

Square roots of negative numbers are not real. The imaginary unit $i$ is defined as $i = \sqrt{-1}$.

• Imaginary Numbers: Numbers involving $i$, such as $3i$ or $-2i$.

• Complex Numbers: Numbers of the form $a + bi$, where $a$ and $b$ are real numbers.

Example: Simplify $\sqrt{-9} = 3i$.

Powers of i

Powers of $i$ repeat in a cycle:

Power	Value
$i^1$	$i$
$i^2$	$-1$
$i^3$	$-i$
$i^4$	$1$

For higher powers, divide the exponent by 4 and use the remainder to determine the value.

Complex Numbers: Operations

Standard Form

Any complex number can be written as $a + bi$.

• Real Part: $a$

• Imaginary Part: $b$

Addition and Subtraction

• Add or subtract real parts and imaginary parts separately.

• Example: $(2 + 4i) + (6 - i) = 8 + 3i$

Multiplication

• Use distributive property (FOIL).

• Remember $i^2 = -1$.

• Example: $(3 + 2i)(1 - 2i) = 3 + 2i - 6i - 4i^2 = 3 - 4(-1) - 4i = 7 - 4i$

Complex Conjugates

• The conjugate of $a + bi$ is $a - bi$.

• Multiplying a complex number by its conjugate yields a real number: $(a + bi)(a - bi) = a^2 + b^2$

Division

• To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator.

• Example: $\frac{3}{1 + 2i}$, multiply by $1 - 2i$:

$\frac{3}{1 + 2i} \times \frac{1 - 2i}{1 - 2i} = \frac{3(1 - 2i)}{(1 + 2i)(1 - 2i)} = \frac{3 - 6i}{1 - 4i^2} = \frac{3 - 6i}{1 + 4} = \frac{3 - 6i}{5}$

Quadratic Equations: Introduction

A quadratic equation is a polynomial equation of degree 2, generally written as $ax^2 + bx + c = 0$.

• Standard form: $ax^2 + bx + c = 0$

• All terms on one side, in descending order of degree.

Example: $2x^2 - 6 = 0$

Factoring Quadratic Equations

Factoring is one method to solve quadratic equations. Set each factor equal to zero and solve for $x$.

• Factor the quadratic expression.

• Set each factor equal to zero.

• Solve for $x$.

Example: $x^2 - 5x + 6 = 0$ factors to $(x - 2)(x - 3) = 0$, so $x = 2$ or $x = 3$.

The Square Root Property

For equations of the form $x^2 = k$, use the square root property:

• $x^2 = k \implies x = \pm \sqrt{k}$

• Solutions may be real or imaginary, depending on $k$.

Example: $x^2 = -9 \implies x = \pm 3i$

Completing the Square

Completing the square is a method to solve any quadratic equation by rewriting it in the form $(x + p)^2 = q$.

• Move constant to one side.

• Add $(\frac{b}{2a})^2$ to both sides.

• Take square root of both sides.

• Solve for $x$.

Example: $x^2 + 6x + 5 = 0$

• Move 5: $x^2 + 6x = -5$

• Add 9: $x^2 + 6x + 9 = 4$

• $(x + 3)^2 = 4$

• $x + 3 = \pm 2 \implies x = -1$ or $x = -5$

The Quadratic Formula

The quadratic formula solves any quadratic equation in standard form:

$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a }$

• Plug in values for $a$, $b$, and $c$.

• Simplify to find solutions.

Example: $x^2 + 4x + 1 = 0$

• $a = 1$, $b = 4$, $c = 1$

• $x = \frac{ -4 \pm \sqrt{ 16 - 4 } }{ 2 } = \frac{ -4 \pm \sqrt{12} }{ 2 } = \frac{ -4 \pm 2\sqrt{3} }{ 2 } = -2 \pm \sqrt{3}$

The Discriminant

The discriminant $D = b^2 - 4ac$ determines the nature and number of solutions to a quadratic equation:

Discriminant ($D$)	Type of Solutions
$D > 0$	Two distinct real solutions
$D = 0$	One real solution (double root)
$D < 0$	Two complex (imaginary) solutions

Choosing a Method to Solve Quadratic Equations

There are several methods to solve quadratic equations. Choose the most efficient based on the equation's form:

Method	When to Use
Factoring	When the quadratic factors easily
Square Root Property	When the equation is in $x^2 = k$ form
Completing the Square	When factoring is difficult or for deriving the quadratic formula
Quadratic Formula	Works for any quadratic equation

Linear Inequalities

Linear inequalities are similar to linear equations but use inequality symbols ($<, >, \leq, \geq$) instead of $=$.

• Interval Notation: A compact way to express solution sets.

• Closed Interval: $[a, b]$ includes endpoints.

• Open Interval: $(a, b)$ excludes endpoints.

• Half-Closed Interval: $[a, b)$ or $(a, b]$ includes one endpoint.

Notation	Set Notation	Interval Notation	Number Line
Closed	$\{ x | a \leq x \leq b \}$	$[a, b]$	Solid dots at $a$ and $b$
Open	$\{ x | a < x < b \}$	$(a, b)$	Open circles at $a$ and $b$
Half-Closed	$\{ x | a \leq x < b \}$	$[a, b)$	Solid dot at $a$, open at $b$

Solving Linear Inequalities:

• Apply the same operations as for equations.

• Important: When multiplying or dividing both sides by a negative number, reverse the inequality sign.

Example: Solve $2x + 12 > 19$

• Subtract 12: $2x > 7$

• Divide by 2: $x > 3.5$

• Interval notation: $(3.5, \infty)$

Linear Inequalities with Fractions and Variables on Both Sides

When inequalities involve fractions or variables on both sides, clear fractions and collect like terms as with equations.

• Express the solution set in interval notation and graph on a number line.

Example: Solve $\frac{1}{2}(x + 1) \geq \frac{1}{2}(2 + 2x)$

• Multiply both sides by 2: $x + 1 \geq 2 + 2x$

• Subtract $x$: $1 \geq 2 + x$

• Subtract 2: $-1 \geq x$ or $x \leq -1$

• Interval notation: $(-\infty, -1]$

Additional info: These notes cover all major subtopics from Ch. 1 of a Precalculus course, including linear and rational equations, complex numbers, quadratic equations (factoring, square root property, completing the square, quadratic formula), and linear inequalities. Tables have been recreated for classification and comparison purposes. All equations are provided in LaTeX format for clarity.