Mastering Linear Equations and Inequalities

This cheat sheet covers the fundamental concepts of lines, including their slopes, equations, and relationships (parallel/perpendicular). It also details solving various types of equations and inequalities, including linear, quadratic, radical, rational, and absolute value forms.

Core Principles

Lines have properties like slope and intercept that define their equation.
Different forms of linear equations (point-slope, slope-intercept, general) can represent the same line.
Parallel lines share the same slope; perpendicular lines have slopes that are negative reciprocals.
Solving equations involves isolating the variable, often requiring manipulation like factoring or using the quadratic formula.
Inequalities are solved similarly to equations, but multiplying/dividing by a negative number reverses the inequality sign.
Absolute value equations and inequalities involve considering both positive and negative cases of the expression.
Extraneous solutions can arise when solving radical equations or when squaring both sides, requiring verification.
Rational inequalities require careful consideration of the signs of the numerator and denominator and the exclusion of values that make the denominator zero.

Action Steps

To find the slope of a line, use the formula $m = \frac{\text{rise}}{\text{run}}$ with two points.
To graph a line, plot at least two points and connect them.
To write a linear equation, determine the slope and y-intercept, or use two points.
When solving equations, isolate the variable by performing inverse operations.
When solving inequalities, perform inverse operations, but reverse the inequality sign when multiplying or dividing by a negative number.
For radical equations, isolate the radical, square both sides, and check for extraneous solutions.
For rational equations/inequalities, find a common denominator, clear fractions, and solve, noting excluded values.
For absolute value equations, split into two cases ($x=c$ and $x=-c$).
For absolute value inequalities, split based on the inequality sign (e.g., $x>c$ or $x<-c$ for gt;$, and $-c \leq x \leq c$ for $\\leq$).
Always check solutions for radical equations and sometimes for rational equations to eliminate extraneous solutions.

Formulas

Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Point-Slope Form: $y - y_1 = m(x - x_1)$
Slope-Intercept Form: $y = mx + b$
General Form: $Ax + By + C = 0$
Perpendicular Slopes: $m_1 m_2 = -1$
Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Absolute Value Equation: $|x| = c \implies x = c \text{ or } x = -c$
Absolute Value Inequality (gt; c$): $|x| > c \implies x > c \text{ or } x < -c$
Absolute Value Inequality ($\\leq c$): $|x| \leq c \implies -c \leq x \leq c$

Key Terms

Slope: The measure of the steepness of a line, calculated as the ratio of the change in y-values to the change in x-values (rise over run).
Y-intercept: The point where a line crosses the y-axis (where x=0).
Point-Slope Form: A form of a linear equation: $y - y_1 = m(x - x_1)$.
Slope-Intercept Form: A form of a linear equation: $y = mx + b$.
General Form: A form of a linear equation: $Ax + By + C = 0$.
Parallel Lines: Lines with the same slope.
Perpendicular Lines: Lines whose slopes are negative reciprocals of each other.
Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$.
Discriminant: The part of the quadratic formula under the square root: $b^2 - 4ac$. It indicates the nature of the roots.
Complex Number: A number in the form $a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$.
Imaginary Unit (i): Defined as $\sqrt{-1}$.
Radical Equation: An equation containing a variable in the radicand.
Rational Equation: An equation containing a variable in the denominator of a fraction.
Inequality: A mathematical statement comparing two expressions using symbols like lt;, >, \leq, \geq$.
Absolute Value: The distance of a number from zero on the number line, always non-negative.
Extraneous Solution: A solution obtained through the solving process that does not satisfy the original equation.

Real World Examples

Calculating the cost of a phone plan based on monthly fees and per-minute charges.: Using inequalities to determine which plan is more advantageous for a given number of minutes.
A group of friends sharing the cost of a vacation home.: Setting up and solving equations to find the number of people in the group based on cost per person.
Modeling projectile motion or other physical phenomena.: Using quadratic equations to find maximum height, time of flight, or other key parameters.

Timeline

Ancient Greece: Early concepts of geometry and lines developed.
17th Century: René Descartes and Pierre de Fermat independently developed analytic geometry, linking algebra and geometry through coordinate systems.
17th-18th Century: Development of calculus by Newton and Leibniz, heavily relying on the concept of slopes and rates of change.
18th Century: Leonhard Euler made significant contributions to algebra, including work on complex numbers.
19th Century: Formalization of complex numbers and their arithmetic.

People

René Descartes: Co-developer of analytic geometry, linking algebraic equations to geometric shapes.
Pierre de Fermat: Co-developer of analytic geometry and made contributions to number theory.
Leonhard Euler: Made foundational contributions to complex numbers and many other areas of mathematics.
Isaac Newton: Developed calculus, a fundamental tool in understanding rates of change and slopes.