📈College Algebra Unit 2 – Equations and Inequalities

Key Concepts and Definitions

• Equation a mathematical statement that two expressions are equal, indicated by the equals sign (=)
• Inequality a mathematical statement comparing two expressions using symbols such as >, <, ≥, or ≤
  • Strict inequalities use > or < and do not include the boundary value
  • Inclusive inequalities use ≥ or ≤ and include the boundary value
• Variable a letter or symbol representing an unknown value in an equation or inequality (x, y, z)
• Coefficient a numerical value multiplied by a variable in an equation or inequality ($3x$, where 3 is the coefficient)
• Constant a fixed value in an equation or inequality that does not change ($y = 2x + 5$, where 5 is the constant)
• Like terms terms in an equation or inequality that have the same variables raised to the same powers ($3x^2$ and $-5x^2$ are like terms)
• Quadratic equation an equation containing a second-degree polynomial, typically in the form $ax^2 + bx + c = 0$

Linear Equations and Their Applications

• Linear equation an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants and $a ≠ 0$
  • $a$ represents the slope or rate of change
  • $b$ represents the y-intercept or starting point
• Slope-intercept form a way to write linear equations, expressed as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept
• Point-slope form another way to write linear equations, expressed as $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope
• Standard form a way to write linear equations, expressed as $ax + by = c$, where $a$, $b$, and $c$ are constants
• Parallel lines lines that never intersect and have the same slope
• Perpendicular lines lines that intersect at a 90-degree angle and have slopes that are negative reciprocals of each other
• Applications of linear equations include modeling real-world situations such as cost analysis, population growth, and distance-rate-time problems

Solving Inequalities

• Solving inequalities involves finding the set of values that satisfy the inequality
• When multiplying or dividing an inequality by a negative number, the direction of the inequality symbol must be reversed
• Graphing inequalities on a number line
  • Use an open circle for strict inequalities (< or >)
  • Use a closed circle for inclusive inequalities (≤ or ≥)
  • Shade the portion of the number line that satisfies the inequality
• Compound inequalities involve connecting two or more inequalities with "and" (∧) or "or" (∨) logical operators
  • "And" inequalities result in a solution set that satisfies both inequalities simultaneously
  • "Or" inequalities result in a solution set that satisfies at least one of the inequalities
• Absolute value inequalities involve inequalities containing absolute value expressions (|x|)
  • Solve by considering two separate cases: one for the positive value and one for the negative value within the absolute value symbols

Systems of Equations

• A system of equations is a set of two or more equations with the same variables
• Solving a system of equations involves finding the values of the variables that satisfy all equations simultaneously
• Substitution method solve one equation for a variable and substitute the resulting expression into the other equation
• Elimination method multiply the equations by constants to eliminate one variable when the equations are added together
• Graphing method graph the equations on the same coordinate plane and find the point(s) of intersection
  • The point(s) of intersection represent the solution(s) to the system
• Systems can have one solution (consistent and independent), no solution (inconsistent), or infinitely many solutions (consistent and dependent)
• Applications of systems of equations include solving problems involving mixtures, money, and rate of work

Quadratic Equations and Functions

• Quadratic equation an equation in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a ≠ 0$
• Quadratic function a function in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a ≠ 0$
  • The graph of a quadratic function is a parabola
• Solving quadratic equations
  • Factoring find two linear factors that, when multiplied, result in the quadratic expression
  • Quadratic formula $x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation
  • Completing the square rewrite the quadratic expression as a perfect square trinomial plus a constant
• Discriminant the expression under the square root in the quadratic formula ($b^2 - 4ac$)
  • Determines the nature of the roots (real and distinct, real and equal, or complex)
• Vertex the point at which a parabola changes direction, either a maximum or minimum point
  • Vertex formula $(\frac{-b}{2a}, f(\frac{-b}{2a}))$, where $a$ and $b$ are the coefficients of the quadratic function

Graphing Techniques

• Coordinate plane a two-dimensional plane formed by the intersection of a horizontal x-axis and a vertical y-axis
• Origin the point of intersection of the x-axis and y-axis, represented as (0, 0)
• Quadrants the four regions of the coordinate plane formed by the x-axis and y-axis (I, II, III, IV)
• Plotting points represent ordered pairs (x, y) as points on the coordinate plane
• Transformations of graphs
  • Vertical shift $f(x) + k$ shifts the graph up by $k$ units if $k > 0$ or down by $|k|$ units if $k < 0$
  • Horizontal shift $f(x - h)$ shifts the graph right by $h$ units if $h > 0$ or left by $|h|$ units if $h < 0$
  • Vertical stretch/compression $af(x)$ stretches the graph vertically by a factor of $|a|$ if $|a| > 1$ or compresses the graph vertically by a factor of $|a|$ if $0 < |a| < 1$
  • Horizontal stretch/compression $f(bx)$ compresses the graph horizontally by a factor of $|b|$ if $|b| > 1$ or stretches the graph horizontally by a factor of $|b|$ if $0 < |b| < 1$
  • Reflection across the x-axis $-f(x)$ reflects the graph across the x-axis
  • Reflection across the y-axis $f(-x)$ reflects the graph across the y-axis

Word Problems and Real-World Applications

• Translating word problems into equations or inequalities
  • Identify the unknown quantity and assign a variable
  • Determine the relationships between the unknown and known quantities
  • Write an equation or inequality that represents the problem
• Distance-rate-time problems involve the relationship $distance = rate × time$
  • Often require setting up a system of equations to solve for unknown distances, rates, or times
• Mixture problems involve combining two or more substances with different concentrations or prices
  • Often require setting up a system of equations to solve for unknown quantities or concentrations
• Work problems involve the relationship between the rate at which a task is completed and the time it takes to complete the task
  • Often require setting up a system of equations to solve for unknown rates or times
• Revenue, cost, and profit problems involve the relationships $revenue = price × quantity$, $profit = revenue - cost$, and $cost = fixed cost + variable cost$
  • May require setting up a quadratic equation to solve for unknown quantities or optimize profit

Common Mistakes and How to Avoid Them

• Forgetting to distribute negative signs when expanding or factoring expressions
  • Double-check each term to ensure the correct sign is applied
• Incorrectly combining unlike terms
  • Only combine terms with the same variables raised to the same powers
• Dividing by zero or taking the square root of a negative number
  • Always check for these situations and consider any restrictions on the variable
• Misinterpreting the direction of inequality symbols when multiplying or dividing by a negative number
  • Remember to reverse the direction of the inequality symbol when multiplying or dividing by a negative number
• Graphing inequalities incorrectly on a number line
  • Use an open circle for strict inequalities and a closed circle for inclusive inequalities
  • Shade the correct portion of the number line that satisfies the inequality
• Failing to check solutions in the original equation, inequality, or word problem
  • Always substitute the solution back into the original problem to verify its correctness
• Misinterpreting the meaning of variables or coefficients in word problems
  • Carefully read the problem and identify the meaning of each variable and coefficient
  • Ensure the equation or inequality accurately represents the problem