College Algebra Review: Inequalities, Expressions, Perimeter, and Conversions

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Inequalities and Interval Notation

Representing Solutions to Inequalities

In algebra, solutions to inequalities can be represented in several ways: on a number line, using set-builder notation, and with interval notation. Understanding these representations is essential for solving and communicating solutions to algebraic problems.

Number Line: Visual representation showing the range of values that satisfy the inequality.
Set-Builder Notation: Describes the set of all numbers that satisfy a given condition, e.g., {x | x < 12}.
Interval Notation: Uses parentheses and brackets to indicate the set of numbers included, e.g., (-∞, 12).

Example Table:

Number Line	Inequality or Set-Builder Notation	Interval Notation
Open circle at 12, arrow left	{x \| x < 12}	(-∞, 12)
Closed circle at 8, open at 14, line between	{x \| 8 ≤ x < 14}	[8, 14)
Open circle at -2, arrow right	{x \| x > -2}	(-2, ∞)
Closed circle at 5	{x \| x = 5}	{5}
Open circles at -3, 2, 7	{x \| x = -3 or x = 2 or x = 7}	{-3, 2, 7}

Algebraic Expressions and Simplification

Operations with Fractions and Variables

Algebraic expressions often require simplification, including combining like terms, distributing, and reducing fractions. Answers should be given in reduced form, preferably as improper fractions when necessary.

Combine Like Terms: Add or subtract terms with the same variable and exponent.
Distributive Property: Multiply each term inside parentheses by the term outside.
Reduce Fractions: Express answers in the form a/b where a and b have no common factors.

Example:

Simplify
Simplify
Simplify

Perimeter and Area Calculations

Finding Perimeter of Polygons

The perimeter of a polygon is the sum of the lengths of its sides. For rectangles, use , where l is length and w is width. For other polygons, add the lengths of all sides.

Pentagon Example: If the sides are 5a, 6a, 8a, 4a, and 3a, then
Rectangle Example: For width and length ,

Translating and Solving Algebraic Expressions

Translating Words to Algebraic Expressions

Translating verbal statements into algebraic expressions is a key skill in algebra. Common phrases include:

The difference of 17 and 3 times a number:
The sum of y and 4:
The product of 5 and s:
The difference of 3 times y squared and y:

Solving Linear Equations

To solve linear equations, isolate the variable using inverse operations. Always check your solution by substituting back into the original equation.

Example: Solve
Example: Solve

Unit Conversions and Scientific Notation

Converting Units

Unit conversions are essential in algebra and science. Use conversion factors to change from one unit to another. For example, , .

Example: Convert 2 kg to grams:
Example: Convert 5 ft to inches:

Scientific Notation

Scientific notation expresses large or small numbers as a product of a number between 1 and 10 and a power of 10. For example, .

Example:
Example:

Graphing and Plotting Points

Plotting Points on the Coordinate Plane

Points are plotted as ordered pairs (x, y) on the coordinate plane. The first value is the x-coordinate (horizontal), and the second is the y-coordinate (vertical).

Example: Plot (3, -2): Move 3 units right and 2 units down from the origin.
Example: Plot (-1, 4): Move 1 unit left and 4 units up from the origin.

Additional info:

Some questions involve perimeter and area, which are foundational for algebraic modeling.
Unit conversions and scientific notation are included, which are common in applied algebra problems.
Problems cover inequalities, equations, expressions, and basic graphing, all core to College Algebra.