Essential Mathematics: Geometry, Scientific Notation, Equations, and Inequalities

Geometric Area Calculations

• Triangle Area Formula:
  • $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
• Calculation Procedure:
  • Identify the base and corresponding perpendicular height (e.g., base = $22\,\text{cm}$, height = $6\,\text{cm}$, ignoring extraneous side lengths such as $14\,\text{cm}$).
  • Substitute into the formula: $\text{Area} = \frac{1}{2} \times 22\,\text{cm} \times 6\,\text{cm} = 11\,\text{cm} \times 6\,\text{cm} = 66\,\text{cm}^2$.

Scientific and Standard Notation

• Scientific Notation Structure:
  • Expressed in the form $a \times 10^n$, where $1 \le |a| < 10$ and $n$ is an integer.
• Converting Standard Form to Scientific Notation:
  • Shift the decimal point to create a coefficient $a$ such that $1 \le |a| < 10$.
  • Moving the decimal point to the right yields a negative exponent ($n < 0$):
  • $0.00856 = 8.56 \times 10^{-3}$ (shifted $3$ places right)
  • $0.000896\,\text{moles per L} = 8.96 \times 10^{-4}\,\text{moles per L}$ (shifted $4$ places right)
  • Moving the decimal point to the left yields a positive exponent ($n > 0$):
  • $17,200,000\,\text{square miles} = 1.72 \times 10^7\,\text{square miles}$ (shifted $7$ places left)
• Converting Scientific Notation to Standard Form:
  • Positive exponent: Shift decimal point right by $n$ places, filling empty positions with zeros:
  • $9.9 \times 10^3\,\text{pounds} = 9900\,\text{pounds}$
  • Negative exponent: Shift decimal point left by $|n|$ places, filling empty positions with leading zeros:
  • $4.1 \times 10^{-2}\,\text{seconds} = 0.041\,\text{seconds}$

Solving Literal and Multi-Step Linear Equations

• Literal Equations:
  • Treat non-target variables as constants and isolate the target variable using inverse operations in reverse order of operations (PEMDAS/BODMAS).
  • Solving $K = 5r - s$ for $r$:
  • Add $s$ to both sides: $K + s = 5r$
  • Divide both sides by $5$: $r = \frac{K + s}{5}$
• Multi-Step Equations with Distribution:
  • Distribute coefficients into parentheses and combine like terms on each side before isolating the variable.
  • Infinite Solutions (Identity):
  • $4(y + 1) - 7 = 3(y - 1) + y$
  • Distribute: $4y + 4 - 7 = 3y - 3 + y$
  • Combine like terms: $4y - 3 = 4y - 3$
  • Subtract $4y$: $-3 = -3$
  • A true numerical statement indicates all real numbers are solutions.
  • Unique Solution:
  • $3(u - 2) - 4u = 2(u - 9)$
  • Distribute: $3u - 6 - 4u = 2u - 18$
  • Combine like terms: $-u - 6 = 2u - 18$
  • Isolate variable and constants: $-6 = 3u - 18 \implies 12 = 3u$
  • Divide by $3$: $u = 4$

Graphing Linear and Compound Inequalities on a Number Line

• Boundary Circle Rules:
  • Open Circle: Used for strict inequalities ($<$ or $>$) to show the boundary value is excluded.
  • Closed Circle: Used for non-strict inequalities ($\le$ or $\ge$) to show the boundary value is included.
• Single Inequalities:
  • $a \le -7$: Closed circle at $-7$, shaded ray extending to the left.
  • $y < -8$: Open circle at $-8$, shaded ray extending to the left.
• Compound "And" Inequalities (Intersection):
  • Requires values to satisfy both conditions simultaneously; represented as a continuous segment between two endpoints.
  • $x > -3 \text{ and } x \le 7$: Open circle at $-3$, closed circle at $7$, connected by a line segment.
  • $x < 7 \text{ and } x \ge 5$ (expressed as $5 \le x < 7$): Interval $[5, 7)$, graphed with a closed circle at $5$, an open circle at $7$, and a connecting line segment.
• Compound "Or" Inequalities (Union):
  • Requires values to satisfy at least one condition; represented as two disjoint rays pointing in opposite directions.
  • $x > 8 \text{ or } x < 1$: Open circle at $1$ with a ray to the left; open circle at $8$ with a ray to the right.
  • $x \le -4 \text{ or } x > 8$: Closed circle at $-4$ with a ray to the left; open circle at $8$ with a ray to the right.