08/31

Linear Equations and Literal Equations

• Standard Form of a Linear Equation:

  • Expression: $\mathbf{ax + b = 0}$

  • Represents a linear equation where all variable terms and constants are placed on the left side and set equal to zero.

• Fundamental Operations for Solving Linear Equations:

  • Add or subtract the exact same real number quantity to/from both sides.

  • Multiply or divide both sides by the exact same non-zero real number quantity.

• Step-by-Step Example 1: Solving a Linear Equation with Fractions:

  • Given Equation: $\frac{x + 2}{2} - \frac{x}{3} = 5$

  • Strategy: Eliminate fraction denominators by multiplying every term across the entire equation by the Least Common Denominator (LCD). For denominators $2$ and $3$, the LCD is $6$.

  • Step 1 (Multiply terms by LCD $6$): $6 \times \left(\frac{x + 2}{2}\right) - 6 \times \left(\frac{x}{3}\right) = 6 \times 5$

  • Step 2 (Simplify coefficients): $3(x + 2) - 2(x) = 30$

  • Step 3 (Distribute terms): $3x + 6 - 2x = 30$

  • Step 4 (Combine like terms): $x + 6 = 30$

  • Step 5 (Isolate $x$): Subtract $6$ from both sides: $x = 24$

  • Verification: Substitute $x = 24$ back into original equation:

  • $\frac{24 + 2}{2} - \frac{24}{3} = 5$

  • $\frac{26}{2} - 8 = 5$

  • $13 - 8 = 5$ (True statement).

• Step-by-Step Example 2: Solving a Literal Equation for a Specified Variable:

  • Given Equation: Solve $m = nt + nr$ for variable $n$

  • Step 1 (Factor out variable $n$ from the right side): $m = n(t + r)$

  • Step 2 (Isolate $n$ by dividing both sides by the factor $(t + r)$): $n = \frac{m}{t + r}$

Linear Inequalities and Interval Notation

• Inequality Operators:

  • Symbols include: Less than ($<$), greater than ($>$), less than or equal to ($\le$), greater than or equal to ($\ge$).

• Formal Definition of "Less Than":

  • $a < b$ indicates that $a$ is located to the left of $b$ on a horizontal real number line.

  • Mathematically, $a < b$ implies there exists a positive real number $p$ such that $a + p = b$

  • Direction Equivalence: $a < b \iff b > a$

• Numerical Inequality Comparison Examples:

  • $4 < 12$

  • $-10 < -3$ (since $-10$ is positioned further left than $-3$ on the number line)

  • $0 > -19$

• Properties of Inequalities:

  • Adding or subtracting any real number on both sides maintains the existing inequality direction.

  • Multiplying or dividing both sides by a positive real number maintains the existing inequality direction.

  • Negative Multiplicative Property: Multiplying or dividing both sides of an inequality by a negative real number requires flipping (reversing) the inequality sign.

  • Proof Demonstration of Sign Reversal:

  • Consider the true statement: $-2 > -9$

  • Multiplying both sides by positive $3$ gives $-6 > -27$ (remains true).

  • Multiplying $-2 > -9$ by negative $-3$ converts terms to positive $6$ and positive $27$. Leaving the sign unchanged yields $6 > 27$ (false). Flipping the sign yields $6 < 27$ (true).

• Representations of Inequality Solutions:

  1. Inequality Notation: Uses comparison symbols ($<, \le, >, \ge$).

  2. Interval Notation:

  • Closed endpoint ($\le$ or $\ge$): Represented by square brackets $[$ or $]$.

  • Open endpoint ($<$ or $>$): Represented by parentheses $($ or $)$.

  • Infinity symbols ($\infty$ or $-\infty$): Always bound by parentheses $($ or $)$.

  1. Line Graph Notation:

  • Place square brackets or parentheses directly on the endpoint numbers on the number line.

  • Shade the valid line segment between endpoints or draw an extending shaded arrow pointing continuously toward $\infty$ or $-\infty$.

• Interval Conversion Practice Exercises:

  • Exercise A: Convert the interval $[-5, 2)$ to inequality notation and line graph.

  • Double Inequality: $-5 \le x < 2$

  • Graph Representation: Square bracket $[$ at $-5$, continuous shading up to $2$, and parenthesis $)$