Study Guide for Linear and Rational Equations

Linear and Rational Equations

• Linear Equations Definition and Structure:

  • A linear equation is an algebraic equation in which the highest exponent of the variable is $1$.
  • The standard form of a single-variable linear equation is typically expressed as $ax + b = 0$, where $a$ and $b$ are real numbers and $a \neq 0$.

• Step-by-Step Procedure to Solve Linear Equations:

  • Step 1: Move all terms involving the variable to one side of the equation using addition or subtraction.
  • Step 2: Simplify the equation by combining like terms on each respective side.
  • Step 3: Isolate the variable completely by applying addition or subtraction first to eliminate constant terms, followed by multiplication or division to clear the coefficient of the variable.

• Rational Equations Definition and Structure:

  • Rational equations are equations containing one or more fractions that feature polynomial expressions in the numerator, denominator, or both.

• Step-by-Step Procedure to Solve Rational Equations:

  • Step 1 (Identify Restrictions): Determine the restrictions on the variable by setting each denominator equal to $0$ and solving. Any value that causes a denominator to become zero is undefined and cannot be part of the solution set.
  • Step 2 (Clear Denominators): Find the least common denominator (LCD) of all rational terms and multiply both sides of the equation by this common denominator to eliminate all fractions.
  • Step 3 (Solve): Solve the resulting simplified linear (or polynomial) equation.
  • Step 4 (Check Extraneous Solutions): Compare all obtained solutions against the domain restrictions identified in Step 1. Any candidate solution that makes any original denominator zero is an extraneous solution and must be discarded.

Simplifying Imaginary Square Roots and Complex Conjugates

• The Imaginary Unit:

  • The imaginary unit is denoted by the symbol $i$.
  • Fundamental definition: $i = \sqrt{-1}$, which establishes the identity:     $i^2 = -1$

• Simplification Procedure for Imaginary Square Roots:

  • Express the radical containing a negative radicand by factoring out $\sqrt{-1}$ as $i$.
  • For any positive real number $k$, $\sqrt{-k} = i\sqrt{k}$.
  • Example:     $\sqrt{-4} = \sqrt{4}\sqrt{-1} = 2i$

• Complex Conjugates Multiplication Property:

  • Complex conjugates are pairs of binomial complex numbers having opposite signs between their real and imaginary parts, appearing in the form $(a + bi)$ and $(a - bi)$.
  • Multiplying complex conjugates eliminates the imaginary unit and yields a purely real sum of squares:     $(a + bi)(a - bi) = a^2 - abi + abi - b^2 i^2 = a^2 - b^2(-1) = a^2 + b^2$

Quadratic Equations, Discriminant Analysis, and Word Problems

• The Quadratic Formula:

  • For any general quadratic equation in standard form $ax^2 + bx + c = 0$ (where $a \neq 0$), the solutions are given by:     $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• The Discriminant ($D$):

  • The radicand within the quadratic formula is designated as the discriminant:     $D = b^2 - 4ac$
  • The value and sign of $D$ determine the nature and number of the roots of the quadratic equation:
  • Case 1 ($D > 0$): The equation yields two distinct real roots.
  • Case 2 ($D = 0$): The equation yields exactly one real root (a repeated or double root), given by $x = -\frac{b}{2a}$.
  • Case 3 ($D < 0$): The equation yields two complex conjugate solutions (no real roots), containing imaginary components.

• Solving Quadratic Word Problems:

  • Identify the unknown quantities and define variable representations based on the given contextual relationships.
  • Set up the quadratic equation adhering to the geometric, physical, or financial constraints outlined in the problem.
  • Solve the resulting quadratic equation using the most efficient method available, such as factoring, completing the square, or using the quadratic formula.
  • Evaluate the solutions within the real-world context of the problem, discarding mathematically valid roots that lack physical meaning (e.g., negative dimensions, negative time, or fractional units where discrete values are required).

Solving Linear, Compound, and Absolute Value Inequalities

• Linear Inequalities:

  • Linear inequalities follow algebraic manipulations identical to linear equations, but relate expressions using order symbols:
  • Greater than: $>$
  • Less than: $<$
  • Greater than or equal to: $\ge$
  • Less than or equal to: $\le$
  • Sign Inversion Rule: Whenever both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality symbol must be reversed.
  • Solutions to linear inequalities represent continuous ranges of real numbers and can be depicted graphically on a number line using open circles (for strict inequalities $<, >$) or closed dots (for inclusive inequalities $\le, \ge$).

• Compound Inequalities:

  • A compound inequality consists of two distinct inequalities joined by the logical connectors "and" or "or".
  • Solving procedure:
  • Solve each constituent inequality independently.
  • For "and" conjunctions (intersections), find the overlapping region satisfying both conditions simultaneously.
  • For "or" disjunctions (unions), combine all solution regions satisfying at least one of the conditions.

• Absolute Value Inequalities:

  • Absolute value inequalities represent distance from zero and must be solved by splitting the problem into two distinct cases for the expression contained within the absolute value bars:
  • Case (a): Evaluate the expression exactly as written without the absolute value bars.
  • Case (b): Negate the algebraic expression (or reverse the inequality sign and negate the opposing boundary constant).
  • Standard forms:
  • For $|u| < c$ (where $c > 0$), the inequality converts to a bounded compound intersection: $-c < u < c$.
  • For $|u| > c$ (where $c > 0$), the inequality converts to a disjoint compound union: $u > c$ or $u < -c$.

Linear Equations and Properties of Lines

• Point-Slope Form:

  • Used when a line passes through a designated coordinate point $(x_1, y_1)$ with a known slope $m$:     $y - y_1 = m(x - x_1)$

• Two Points Form:

  • Used when two distinct coordinate points $(x_1, y_1)$ and $(x_2, y_2)$ on the line are known.
  • The slope is first determined by the rise-over-run ratio:     $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • Incorporating the slope definition yields the full equation:     $y - y_1 = \left(\frac{y_2 - y_1}{x_2 - x_1}\right)(x - x_1)$

• Slope-Intercept Form:

  • Expresses the linear relation explicitly in terms of the slope $m$ and vertical intercept $b$:     $y = mx + b$
  • The parameter $m$ represents the steepness and rate of change, while $b$ represents the $y$-coordinate of the $y$-intercept at $(0, b)$.

• Parallel and Perpendicular Line Relationships:

  • Parallel Lines: Lines in the same plane that never intersect share identical rates of change:     $m_1 = m_2$
  • Perpendicular Lines: Lines that intersect at right angles ($90^\circ$) exhibit slopes that are opposite (negative) reciprocals of each other:     $m_1 \cdot m_2 = -1$$m_2 = -\frac{1}{m_1}$

Functions, Function Evaluation, and Domain Restrictions

• Evaluating Functions:

  • To evaluate a function $f(x)$ at a specified input value, substitute the input numerical value or algebraic expression in place of every occurrence of the independent variable $x$ within the function rule, then simplify completely.

• Piecewise-Defined Functions:

  • Functions defined by multiple sub-functions, where each specific formula applies only across a designated domain sub-interval.
  • Evaluation procedure: Determine which sub-interval condition the given input value satisfies, and evaluate the input exclusively within the corresponding formula piece.

• Domain of Functions:

  • The domain is the complete set of all allowable real inputs ($x$-values) that result in real, defined output values.
  • Key analytical restrictions to identify:
  • Rational Expressions: Denominators cannot equal zero. Identify and exclude values of $x$ that solve $\text{denominator} = 0$.
  • Radical Expressions: Radicands of even-degree roots (such as square roots) cannot be negative. Enforce the condition $\text{radicand} \ge 0$ and solve for $x$.

Cost, Revenue, Profit Functions, and Break-Even Analysis

• Business Function Definitions:

  • Cost Function ($C$ or $C(x)$): The total financial expense required to produce $x$ units of a product, typically composed of fixed overhead costs plus variable production costs per unit.
  • Revenue Function ($R$ or $R(x)$): The total monetary income generated from selling $x$ units, generally calculated as unit price multiplied by quantity sold.
  • Profit Function ($P$ or $P(x)$): The net earnings remaining after total production costs are subtracted from total sales revenue:     $P(x) = R(x) - C(x)$

• Break-Even Analysis:

  • Break-even occurs when total revenue equals total cost, resulting in zero net profit ($P(x) = 0$).
  • Break-even points are determined algebraically by finding the intersection of the cost and revenue curves:     $C = R$

Parabolas and Quadratic Analysis

• Vertex Form of a Parabola:

  • The equation of a quadratic function transformed to explicitly display its turning point (vertex) $(h, k)$:     $y = a(x - h)^2 + k$

• Vertex Coordinates from Standard Form ($y = ax^2 + bx + c$):

  • The horizontal coordinate $h$ lies along the axis of symmetry:     $h = -\frac{b}{2a}$
  • The vertical coordinate $k$ is obtained by evaluating the function at the symmetry axis:     $k = f(h) = f\left(-\frac{b}{2a}\right)$

• Finding Horizontal ($x$-) Intercepts:

  • Set the dependent variable equal to zero:     $y = 0$
  • Solve the resulting quadratic equation $ax^2 + bx + c = 0$ using factoring, completing the square, or the quadratic formula.

• Maximization and Minimization of Quadratic Functions:

  • The leading coefficient $a$ determines the vertical concavity and orientation of the parabola:
  • If $a > 0$, the parabola opens upward, and the vertex represents the absolute minimum value of the function.
  • If $a < 0$, the parabola opens downward, and the vertex represents the absolute maximum value of the function.
  • For optimization applications (such as finding maximum revenue, maximum profit, or minimum cost), the extremum occurs at $x = h$, and the extreme value achieved is the vertical coordinate $k$.