A variable is typically a lowercase letter used to represent an unknown quantity or a placeholder. Common variables used in algebra are x, y, and z. For instance, in the example of Mark earning $63 per day plus tips, if 'x' represents the unknown tips, his daily earnings are modeled as $63 + x. Changing the value of 'x' changes his total earnings.

A term is a single number, variable, or a number multiplied by one or many variables (e.g., 4x, 9y, 24xyz). A coefficient is the numerical part of a term that multiplies the variable(s). A constant is a number that stands alone, meaning its value does not change, unlike variables.

A ratio compares two quantities (e.g., 12 boys to 3 girls, simplified as 4:1). A rate is a ratio where units differ (e.g., miles per gallon). A proportion states that two ratios are equal and can be solved using cross-multiplication. This sets up an equation (e.g., a/b = c/d becomes ad = bc), which can then be solved for unknown variables.

Inequalities involve symbols like greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). These often have a range of values as solutions, expressed using interval notation (e.g., (2, ∞) for x > 2) and graphed on a number line (parentheses for strict inequalities, brackets for non-strict). The addition property of inequality allows adding or subtracting any number from both sides without changing the solution. However, multiplying or dividing both sides by a negative number requires reversing the inequality direction.

For inequalities with fractions, multiply by the LCD to clear them. For multiple steps, simplify each side, isolate the variable term using addition/subtraction, and then the variable using multiplication/division. Remember to flip the inequality sign if multiplying or dividing by a negative. Three-part inequalities (e.g., 5 < x < 7) are solved by applying operations to all three parts simultaneously, with solutions represented by an interval between two values.

Linear equations in two variables (e.g., ax + by = c) have infinite solutions, typically written as ordered pairs (x, y). To check a solution, substitute the x and y values into the equation. To find solutions, substitute a value for one variable and solve for the other. This process generates ordered pairs that satisfy the equation.

The Cartesian coordinate system (or coordinate plane) consists of a horizontal x-axis and a vertical y-axis, intersecting at the origin (0,0). Points are plotted using ordered pairs (x, y), where 'x' indicates horizontal movement and 'y' vertical. The plane is divided into four quadrants, with specific sign combinations for x and y values in each. Plotting involves finding the x- and y-coordinates and marking their intersection.

Graphing a linear equation in two variables produces a straight line, representing all its infinite solutions. Points can be found using a table of values (e.g., picking x values and solving for y) or the intercept method (setting x=0 to find the y-intercept, and y=0 for the x-intercept). Special cases include vertical lines (x = constant) and horizontal lines (y = constant).

The slope (m) measures a line's steepness, comparing the change in y values (rise) to the change in x values (run). Using the slope formula, m = (y2 - y1) / (x2 - x1), or by visually counting rise over run on a graph, the slope can be calculated. A positive slope indicates the line rises to the right, a negative slope indicates it falls, a horizontal line has a zero slope, and a vertical line has an undefined slope.

Slope-intercept form (y = mx + b) directly reveals the slope (m) and y-intercept (b). Point-slope form [y - y1 = m(x - x1)] is useful when one point (x1, y1) and the slope (m) are known. Standard form (ax + by = c) expresses a linear equation where a, b, and c are real numbers. Converting between forms allows for clarity and efficient graphing.

Parallel lines never intersect and have identical slopes but different y-intercepts. Perpendicular lines intersect at a 90-degree angle, and the product of their slopes is -1. To determine if lines are parallel or perpendicular, convert their equations to slope-intercept form to compare their slopes.

Graphing inequalities involves drawing a boundary line (replacing the inequality symbol with an equals sign). For strict inequalities (>, <), use a dashed line; for non-strict (≥, ≤), use a solid line. To determine the solution region, shade above the line for '>' or '≥' and below for '<' or '≤' (after solving for y). This visual representation shows all ordered pairs that satisfy the inequality.

A system of linear equations involves two or more equations with the same variables. The solution is an ordered pair that satisfies all equations simultaneously. Graphing involves plotting each equation's line; their intersection point is the solution. Special cases include parallel lines (no solution) and identical lines (infinite solutions).

The substitution method solves systems algebraically. Isolate one variable in one equation (preferably one with a coefficient of 1 or -1). Substitute that expression into the other equation, creating a single-variable equation. Solve for that variable, then substitute the result back into either original equation to find the other variable. Always check the solution in both original equations. Special cases (no solution or infinite solutions) result in false or true statements after substitution, respectively.

The elimination method involves manipulating equations so that one variable's coefficients are opposites in both equations. Add the equations vertically, eliminating one variable. Solve for the remaining variable, then substitute its value into either original equation to find the other. The initial step is often to ensure both equations are in standard form (Ax + By = C). If no opposite coefficients exist, multiply one or both equations by a constant to create them. Always verify the solution in both original equations.

Solving word problems with linear systems involves: careful reading, assigning two variables to two unknowns, creating two equations based on the problem's information, solving the system (using substitution or elimination), and stating the answer in context. Examples include 'money' problems involving different item costs and 'motion' problems using the distance formula (D = R * T) to relate upstream/downstream travel.

Exponents (e.g., x^6) indicate repeated multiplication of a base (x) by itself. Rules include the product rule (x^a * x^b = x^(a+b)) and power rules [(x^a)^b = x^(a*b)]. The Greatest Common Factor (GCF) is the largest factor common to all terms in a polynomial. Factoring out the GCF is the reverse of the distributive property, simplifying polynomials by extracting the common factor.

Multiplying polynomials involves distributing each term of one polynomial to every term of the other. For binomials, the FOIL method (First, Outer, Inner, Last) is a shortcut. For more terms, it's a systematic application of the distributive property. Special products like (x + y)² = x² + 2xy + y² and (x - y)² = x² - 2xy + y² offer quicker calculation. When multiplying three or more binomials, multiply two first, then multiply the result by the next. Always combine like terms at the end.

To divide a polynomial by a monomial, express the division as a fraction and then divide each term of the polynomial by the monomial. Simplify each resulting fraction using exponent rules. The accuracy of the result can be verified by multiplying the quotient by the divisor, which should yield the original dividend.

For dividing a polynomial by a non-monomial (e.g., a binomial), use long division, similar to how it's done with numbers. Ensure both polynomials are in standard form, using zero as a placeholder for any missing terms. The process involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the result by the entire divisor, subtracting, bringing down the next term, and repeating until a remainder is found. The remainder is expressed as a fraction over the divisor.

To factor trinomials of the form ax² + bx + c where a=1, find two integers whose sum is 'b' (the coefficient of the x-term) and whose product is 'c' (the constant term). The trinomial then factors into (x + integer1)(x + integer2). If no such integers exist, the polynomial is prime.

When the leading coefficient (a) is not 1, the factoring process is more involved. First, factor out any common GCF. Then, for the remaining trinomial, find two integers whose product is a*c and whose sum is b. Use these integers to rewrite the middle term (bx) as two separate terms, converting the trinomial into a four-term polynomial. Finally, use factoring by grouping to find the binomial factors. Alternatively, the 'reverse FOIL' method (trial and error) can be used, though it is often more tedious.

Special factoring rules offer shortcuts for specific polynomial forms. The 'difference of two squares' (x² - y²) factors into (x + y)(x - y). 'Perfect square trinomials' (x² + 2xy + y² or x² - 2xy + y²) factor into (x + y)² or (x - y)². There are also rules for the 'sum or difference of cubes,' (x³ + y³ = (x+y)(x²-xy+y²) and x³-y³ = (x-y)(x²+xy+y²)), which are crucial for simplifying complex expressions and solving higher-degree equations.

A quadratic equation (ax² + bx + c = 0) involves a squared variable and no higher powers. To solve by factoring, first ensure the equation is in standard form (set equal to zero). Factor the quadratic expression into its binomial factors. Then, use the zero product property, setting each factor equal to zero and solving for the variable. This yields two potential solutions, both of which should be checked in the original equation to confirm validity.

Complex fractions contain fractions in their numerator, denominator, or both. They can be simplified by either simplifying the numerator and denominator separately before performing the main division, or by multiplying the numerator and denominator of the complex fraction by the Least Common Denominator (LCD) of all individual fractions within it. The LCD method often leads to quicker simplification, especially with rational expressions involving variables. When dealing with radical expressions, ensure the radicand contains no fractions, and no radicals exist in the denominator. This involves rationalizing the denominator by multiplying by the conjugate if sums/differences of radicals are present.

To solve equations containing rational expressions, clear the denominators by multiplying both sides of the equation by the LCD of all rational expressions. This transforms the equation into a simpler polynomial equation. Solve the resulting equation (it may be linear or quadratic) and then **critically check all potential solutions in the original equation.** Any solution that makes a denominator zero in the original equation must be rejected, as division by zero is undefined.

Rational expressions are commonly used in motion problems (D = R * T, where D/R = T or D/T = R) and work-rate problems. For motion problems involving currents or winds, speeds are adjusted (e.g., boat speed + current speed downstream). Work-rate problems combine individual work rates (e.g., 1/time_A + 1/time_B = 1/time_together). These problems often lead to rational equations that can be solved by finding the LCD and clearing denominators.

Direct variation describes a relationship where y varies directly with x if y = kx, where 'k' is the constant of variation. As x increases, y increases (if k>0), and vice versa. This is similar to the slope-intercept form (y = mx + b) when b=0, with 'k' acting as the slope. Direct variation can also occur with powers of x (y = kxⁿ), meaning y varies directly with the nth power of x. Problems typically involve finding 'k' from an initial scenario and then using it to solve for an unknown in a new scenario.

Inverse variation describes a relationship where y varies inversely with x if y = k/x, 'k' again being the constant of variation. Unlike direct variation, as x increases, y decreases (if k>0), and vice versa. Similar to direct variation, inverse variation can involve powers of x (y = k/xⁿ). Problems are solved by using an initial data pair to find 'k,' then applying 'k' to solve for the unknown in a second scenario.

An algebraic expression consists of one or more terms separated by plus or minus symbols. The value of an algebraic expression changes depending on the values assigned to its variables. To evaluate, substitute the given variable value and follow the order of operations. The distributive property can be used to simplify expressions with parentheses.

Like terms are terms with identical variable parts, including their exponents. Only coefficients can differ. When combining like terms, the variable part remains unchanged; only the coefficients are added or subtracted. For example, 2x + 3x = 5x, but 2x + 3y cannot be combined as they are not like terms.

An equation is a statement that two algebraic expressions are equal, always including an equality symbol. Unlike expressions, which can be simplified or evaluated, equations seek specific variable values that make the statement true. A solution to an equation is any value that, when substituted for the variable, makes both sides of the equation equal.

The addition property of equality states that adding or subtracting the same number from both sides of an equation does not change its solution. This property is used to isolate the variable in simple equations, often by introducing the opposite of a constant term to make it zero (e.g., to solve x + 4 = 10, subtract 4 from both sides).

The multiplication property of equality allows multiplying or dividing both sides of an equation by the same non-zero number without altering its solution. This is crucial for isolating variables multiplied by coefficients. If the coefficient is a fraction, multiply by its reciprocal. If it's an integer, divide by the integer.

Solving linear equations involves three steps: first, simplify both sides completely using the distributive property and combining like terms; second, isolate the variable term using the addition property of equality; third, isolate the variable using the multiplication property of equality. Always check the solution in the original equation.

To solve equations with fractions, multiply both sides by the least common denominator (LCD) of all fractions to eliminate them. For decimals, identify the largest number of decimal places in any term and multiply both sides by a power of 10 sufficient to clear all decimals. This simplifies the equation for easier solving.

Conditional equations have a specific solution that makes them true. Identities are always true for any variable value, resulting in infinite solutions (e.g., 3x - 12 = 3(x - 4) simplifies to 3x - 12 = 3x - 12). Contradictions are never true, indicated by a false statement after simplification (e.g., -2 = 2), meaning no solution exists.

Solving word problems involves reading carefully, assigning variables to unknowns, writing an equation based on problem information, solving the equation, stating the answer in context (e.g., in a sentence), and checking for reasonableness. Examples include ‘sums of quantities’ problems (e.g., finding individual amounts when the total sum is known) and distance-rate-time problems.

Roots are the reverse of exponentiation. A square root (√) finds a number that, when squared, yields the radicand (the number under the radical symbol). For example, √9 = 3 because 3² = 9. There's a principal (positive) square root and a negative square root (-√9 = -3). Higher roots, like cube roots (³√), seek numbers that, when multiplied by themselves 'n' times (where 'n' is the index), yield the radicand. An even index cannot have a negative radicand and result in a real number, while an odd index can. Perfect squares/cubes are numbers whose roots are rational. Irrational numbers (like √2) have non-repeating, non-terminating decimals.

The product rule for radicals states that √a * √b = √ab, allowing for combining or separating radicals. Simplification means no perfect squares (or cubes, etc.) remain as factors in the radicand, no fractions in the radicand, and no radicals in the denominator. The quotient rule for radicals states √(a/b) = √a / √b. For simplification, factor out perfect squares/cubes from under the radical. When rationalizing a denominator with a single radical (e.g., 2/√7), multiply numerator and denominator by that radical. For complex denominators involving two terms with radicals (e.g., 6 - √5), multiply by its conjugate (6 + √5) to eliminate the radical.