6.1 Algebraic Equations, Inequalities & Coordinate Geometry

Algebraic Equations, Inequalities & Coordinate Geometry

Algebraic reasoning and coordinate geometry form a major component of the FTCE General Knowledge Mathematics Subtest. This domain evaluates an educator's ability to manipulate algebraic expressions, solve linear equations and multi-step inequalities, interpret systems of linear equations, analyze functions, and apply coordinate geometry principles to solve geometric and real-world problems.

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1. Solving Linear Equations in One Variable

A linear equation in one variable is an algebraic statement in which the highest exponent of the variable is 1. The general standard form is: $ax + b = c$ where $a$, $b$, and $c$ are real numbers and $a \neq 0$. Solving a linear equation means isolating the variable through balanced inverse operations.

Properties of Equality

To maintain equality, any mathematical operation performed on one side of an equation must be applied identically to the other side:

• Addition Property: If $a = b$, then $a + c = b + c$.
• Subtraction Property: If $a = b$, then $a - c = b - c$.
• Multiplication Property: If $a = b$, then $a \cdot c = b \cdot c$.
• Division Property: If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.
• Distributive Property: $a(b + c) = ab + ac$.

Systematic Multi-Step Equation Solving Protocol

When encountering complex linear equations with grouping symbols, decimals, or fractions, execute the following standardized sequence:

1. Clear Fractions / Decimals (Optional but Recommended): Multiply all terms on both sides of the equation by the Least Common Denominator (LCD) of all rational fractions, or by powers of 10 for decimals.
2. Apply the Distributive Property: Eliminate all parentheses and brackets by expanding expressions: $a(bx + c) = abx + ac$.
3. Combine Like Terms on Each Side: Group and sum constant terms and variable terms separately on the left side and right side.
4. Collect Variable Terms on One Side: Use addition or subtraction to shift all variable terms to one side of the equation (preferably the side yielding a positive coefficient).
5. Collect Constant Terms on the Opposite Side: Use addition or subtraction to move all numerical constants to the opposite side.
6. Isolate the Variable: Multiply or divide by the variable's numerical coefficient.
7. Verify Solution: Substitute the resulting value back into the original equation to ensure both sides evaluate to identical values.

Special Equation Classifications

Not every linear equation yields a single unique numerical solution:

• Conditional Equation: True for exactly one value of the variable (e.g., $2x + 4 = 10 \implies x = 3$). One unique solution.
• Identity Equation: True for all real numbers (e.g., $3(x + 2) = 3x + 6 \implies 6 = 6$). The solution set is all real numbers ($\mathbb{R}$).
• Inconsistent Equation (Contradiction): False for all values (e.g., $2x + 5 = 2x - 1 \implies 5 = -1$). There is no solution (the empty set $\emptyset$).

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2. Solving & Graphing Linear Inequalities

A linear inequality relates two expressions using inequality symbols: $<$ (less than), $\le$ (less than or equal to), $>$ (greater than), or $\ge$ (greater than or equal to).

The Fundamental Rule for Inequalities

CRITICAL EXAM RULE:
When you multiply or divide both sides of an inequality by a negative number, you MUST reverse (flip) the direction of the inequality sign.

Why the Reversal is Mathematically Mandatory:

Consider the true numerical inequality: $2 < 5$ Multiply both sides by $-1$: $-2 \text{ vs. } -5$ On a number line, $-2$ lies to the right of $-5$, meaning $-2 > -5$. Failure to reverse the inequality symbol produces a mathematically false statement.

+-------------------------------------------------------------------------+
|                  INEQUALITY SYMBOLS & GRAPHING RULES                    |
+-------------------+----------------+------------------+-----------------+
| Symbol            | Meaning        | Circle on Line   | Interval Bound  |
+-------------------+----------------+------------------+-----------------+
| <                 | Strictly less  | Open Circle (○)  | Parenthesis (   |
| >                 | Strictly great | Open Circle (○)  | Parenthesis )   |
| <=                | Less or equal  | Solid Dot (●)    | Bracket [       |
| >=                | Great or equal | Solid Dot (●)    | Bracket ]       |
+-------------------+----------------+------------------+-----------------+```

### Compound Inequalities
- **Conjunction ("AND"):** Satisfies two conditions simultaneously. Represents the **intersection** of solution sets (e.g., $-3 \le 2x + 1 < 7$).
  $$\begin{aligned}
  -3 &\le 2x + 1 < 7 \\
  -4 &\le 2x < 6 \\
  -2 &\le x < 3 \quad \implies \quad [-2, 3)
  \end{aligned}$$
- **Disjunction ("OR"):** Satisfies at least one of the conditions. Represents the **union** of solution sets (e.g., $x + 4 < 1 \text{ or } 2x - 3 \ge 5 \implies x < -3 \text{ or } x \ge 4$).

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## 3. Systems of Linear Equations in Two Variables

A **system of linear equations** consists of two or more equations sharing the same variables:
$$\begin{cases}
a_1 x + b_1 y = c_1 \\
a_2 x + b_2 y = c_2
\end{cases}$$

### Algebraic Solution Methods

#### 1. The Substitution Method
Best utilized when at least one variable has a coefficient of $+1$ or $-1$:
1. Solve one equation for one variable in terms of the other (e.g., $y = 5 - 2x$).
2. Substitute that expression into the other equation.
3. Solve the resulting single-variable equation for $x$.
4. Back-substitute the numerical value of $x$ into the isolated equation to calculate $y$.
5. State the solution as an ordered pair $(x, y)$.

#### 2. The Elimination (Linear Combination) Method
Best utilized when equations are formatted in standard form ($Ax + By = C$):
1. Multiply one or both equations by non-zero constants so that the coefficients of one variable become exact opposites (e.g., $+6y$ and $-6y$).
2. Add the two equations vertically to eliminate that variable.
3. Solve the resulting one-variable equation.
4. Back-substitute into either original equation to determine the remaining variable.

### System Classifications & Geometric Interpretations

+-------------------------------------------------------------------------+ | SYSTEM OF LINEAR EQUATIONS TAXONOMY | +-------------------+--------------------+-----------------+--------------+ | Classification | Slopes & Intercepts| Graphs | Solutions | +-------------------+--------------------+-----------------+--------------+ | Consistent & | Distinct Slopes | Two lines | Exactly One | | Independent | (m1 != m2) | intersect at | Solution | | | | single point | (x, y) | +-------------------+--------------------+-----------------+--------------+ | Inconsistent | Equal Slopes, | Parallel lines | No Solution | | | Different y-ints | that never | (empty set) | | | (m1 = m2, b1 != b2)| intersect | | +-------------------+--------------------+-----------------+--------------+ | Consistent & | Equal Slopes, | Coincident | Infinitely | | Dependent | Equal y-ints | lines (same | Many | | | (m1 = m2, b1 = b2) | exact line) | Solutions | +-------------------+--------------------+-----------------+--------------+```

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4. Coordinate Geometry & Linear Equations

The Cartesian Coordinate Plane is formed by the perpendicular intersection of the horizontal $x$-axis and vertical $y$-axis at the origin $(0, 0)$.

Slope of a Line

The slope ($m$) quantifies the steepness and direction of a line, defined as the ratio of vertical change (rise) to horizontal change (run): $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

• Positive Slope ($m > 0$): Line rises from left to right.
• Negative Slope ($m < 0$): Line falls from left to right.
• Zero Slope ($m = 0$): Horizontal line ($y = c$).
• Undefined Slope: Vertical line ($x = k$, division by zero $\Delta x = 0$).

Key Forms of Linear Equations

1. Slope-Intercept Form: $y = mx + b$ where $m$ is the slope and $(0, b)$ is the y-intercept.
2. Point-Slope Form: $y - y_1 = m(x - x_1)$ where $m$ is the slope and $(x_1, y_1)$ is a known point on the line.
3. Standard Form: $Ax + By = C$ where $A, B, C$ are integers, $A \ge 0$, and $A, B$ are not both zero. The slope is $m = -\frac{A}{B}$, and the y-intercept is $(0, \frac{C}{B})$.

Parallel and Perpendicular Lines

• Parallel Lines: Two non-vertical lines are parallel if and only if their slopes are strictly equal and their y-intercepts are different: $m_1 = m_2 \quad \text{and} \quad b_1 \neq b_2$
• Perpendicular Lines: Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals (their product equals $-1$): $m_1 \cdot m_2 = -1 \iff m_2 = -\frac{1}{m_1}$

Distance and Midpoint Formulas

Given two coordinate points $A(x_1, y_1)$ and $B(x_2, y_2)$:

1. Distance Formula

Derived directly from the Pythagorean theorem: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

2. Midpoint Formula

Calculates the arithmetic average of the coordinates to find the exact center of segment $AB$: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

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5. Functions, Relations & Domain/Range

A relation is any set of ordered pairs $(x, y)$. A function is a specialized relation where each input value ($x$) in the domain maps to exactly one output value ($y$) in the range.

Testing for Functions

• Set / Mapping Test: No input $x$-value may be paired with two different $y$-values. For example, ${(1, 4), (2, 5), (1, 6)}$ is NOT a function because input $1$ maps to both $4$ and $6$.
• Vertical Line Test (Graphical): If any vertical line passes through more than one point on a graph, the graph does NOT represent a function.

Function Notation & Evaluation

The notation $f(x)$ (read "$f$ of $x$") represents the output of function $f$ when evaluated at input $x$.

• To find $f(-3)$ for $f(x) = 2x^2 - 4x + 1$: $f(-3) = 2(-3)^2 - 4(-3) + 1 = 2(9) + 12 + 1 = 18 + 12 + 1 = 31$

Domain Restrictions

Unless otherwise specified, the domain consists of all real numbers except:

1. Rational Denominators Cannot Equal Zero: In $f(x) = \frac{2x + 1}{x - 5}$, the domain excludes $x = 5$ ($x \neq 5$).
2. Even Roots of Negative Numbers are Non-Real: In $g(x) = \sqrt{x - 4}$, the domain requires $x - 4 \ge 0 \implies x \ge 4$.

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Realistic FTCE Mathematical Worked Examples

Worked Example 1: Multi-Step Inequality with Sign Reversal

Problem: Solve the linear inequality: $-4(2x - 3) + 5 > 3x + 28$

Step-by-Step Solution:

1. Apply the distributive property on the left side: $-8x + 12 + 5 > 3x + 28 \implies -8x + 17 > 3x + 28$
2. Subtract $3x$ from both sides: $-11x + 17 > 28$
3. Subtract $17$ from both sides: $-11x > 11$
4. Divide both sides by $-11$ and reverse the inequality sign: $x < \frac{11}{-11} \implies x < -1$
5. Conclusion: The solution set is $x < -1$, or in interval notation, $(-\infty, -1)$.

Worked Example 2: Perpendicular Line Equation

Problem: Find the equation of the line that passes through point $(6, -2)$ and is perpendicular to the line $3x - 4y = 12$.

Step-by-Step Solution:

1. Determine the slope of the given line by converting to slope-intercept form: $-4y = -3x + 12 \implies y = \frac{3}{4}x - 3$ The given slope is $m_1 = \frac{3}{4}$.
2. Determine the perpendicular slope ($m_2 = -\frac{1}{m_1}$): $m_2 = -\frac{4}{3}$
3. Use point-slope form with point $(x_1, y_1) = (6, -2)$ and slope $m = -\frac{4}{3}$: y - (-2) &= -\frac{4}{3}(x - 6) \\ y + 2 &= -\frac{4}{3}x + 8 \\ y &= -\frac{4}{3}x + 6 \end{aligned}$$
4. Conclusion: The equation of the perpendicular line is $y = -\frac{4}{3}x + 6$.