Equations and Inequalities: College Algebra Quiz Study Guide

Equations and Inequalities

Solving Equations

Equations are mathematical statements that assert the equality of two expressions. Solving equations is a fundamental skill in algebra, involving finding the value(s) of the variable(s) that make the equation true.

• Linear Equations: Equations of the form $ax + b = 0$, where $a$ and $b$ are constants.

• Rational Equations: Equations involving fractions whose numerators and/or denominators contain variables.

• Absolute Value Equations: Equations that contain absolute value expressions, such as $|x| = a$.

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

Solution: $2x = 4 \implies x = 2$

Solving for Variables in Denominators

When a variable appears in the denominator, it is important to determine the values that make the denominator zero, as division by zero is undefined.

• Set the denominator equal to zero and solve for the variable.

• These values are excluded from the solution set (called restrictions).

Example: For $\frac{1}{x-2}$, $x-2 = 0 \implies x = 2$ is not allowed.

Word Problems: Area of Rectangles

Many algebra problems involve translating a word problem into an equation. For rectangles, the area is given by:

$A = l \times w$

• l = length

• w = width

When one dimension is described in terms of the other (e.g., length is 5 feet longer than width), substitute and solve for the unknowns.

Example: If area is 60 sq ft and length is 5 ft longer than width: $w(w+5) = 60$

Solving Inequalities

Inequalities compare two expressions using symbols such as $<, >, \leq, \geq$. Solving inequalities involves finding all values of the variable that make the inequality true.

• Linear Inequalities: Similar to equations, but the solution is often a range of values.

• Absolute Value Inequalities: Involve expressions like $|x| < a$ or $|x| > a$.

Example: Solve $2x - 3 < 7$.

Solution: $2x < 10 \implies x < 5$

Absolute Value Inequalities

To solve inequalities involving absolute values:

• $|x| < a$ is equivalent to $-a < x < a$

• $|x| > a$ is equivalent to $x < -a$ or $x > a$

Example: Solve $|x-2| < 5$.

Solution: $-5 < x-2 < 5 \implies -3 < x < 7$

Types of Equations

• Linear Equation: An equation of degree one (e.g., $2x + 3 = 7$).

• Quadratic Equation: An equation of degree two (e.g., $x^2 - 4 = 0$).

• Rational Equation: Contains variables in the denominator (e.g., $\frac{1}{x} = 2$).

Example: $x^2 - 4 = 0$ is a quadratic equation.

Summary Table: Equation Types and Solution Methods

Type	General Form	Solution Method
Linear	$ax + b = 0$	Isolate $x$
Quadratic	$ax^2 + bx + c = 0$	Factor, complete the square, or use quadratic formula
Rational	$\frac{P(x)}{Q(x)} = 0$	Set numerator to zero, exclude values making denominator zero
Absolute Value	$|x| = a$	Write two equations: $x = a$ and $x = -a$

Additional info: These topics are foundational for all subsequent algebraic problem-solving, including systems of equations, functions, and graphing.