5.2 Linear Equations, Quadratic Equations, and Inequalities

5.2 Linear Equations, Quadratic Equations, and Inequalities

Core Principle: Algebraic equations and inequalities constitute approximately 40% to 50% of all Mathematics Knowledge items on the CAT-ASVAB. Mastery of algebraic structures—from solving fractional linear equations and compound inequalities to computing simultaneous $2 \times 2$ systems and solving quadratics via factoring or the quadratic formula—is the single most decisive factor in achieving a top-tier MK score.

---

1. Multi-Step Linear Equations & Literal Equation Rearrangement

To solve a single-variable linear equation, systematically isolate the target variable using inverse operations in reverse operational order:

+-----------------------------------------------------------------------------------------+
|                         5-STEP LINEAR EQUATION SOLVER ALGORITHM                         |
+-----------------------------------------------------------------------------------------+
| Step 1: Clear Fractions/Decimals | Multiply every term on both sides by the LCD.        |
| Step 2: Distribute               | Expand all parentheses: a(b + c) = ab + ac.          |
| Step 3: Combine Like Terms       | Simplify algebraic expressions on each side.         |
| Step 4: Isolate Variable Terms   | Add/subtract variable terms to gather on one side.   |
| Step 5: Isolate Constant & Solve | Divide by the numerical coefficient.                 |
+-----------------------------------------------------------------------------------------+

Step-by-Step Worked Example (Fraction Clearing):

$$\text{Solve for } x: \quad \frac{3x - 1}{4} - \frac{2x + 3}{5} = \frac{1}{2}$$

1. Multiply every term by the $\text{LCD} = 20$: $$20 \cdot \left(\frac{3x - 1}{4}\right) - 20 \cdot \left(\frac{2x + 3}{5}\right) = 20 \cdot \left(\frac{1}{2}\right)$$
2. Simplify multipliers: $$5(3x - 1) - 4(2x + 3) = 10$$
3. Distribute carefully (watching negative signs): $$15x - 5 - 8x - 12 = 10$$
4. Combine like terms: $$7x - 17 = 10$$
5. Add 17 to both sides and divide by 7: $$7x = 27 \implies x = \frac{27}{7}$$

Literal Equations (Formula Rearrangement)

ASVAB items regularly ask you to solve a multi-variable physical formula for a specified variable.

• Example 1: Solve Fahrenheit to Celsius formula for $F$: $$C = \frac{5}{9}(F - 32) \implies \frac{9}{5}C = F - 32 \implies F = \frac{9}{5}C + 32$$
• Example 2: Solve Cylinder Volume for height $h$: $$V = \pi r^2 h \implies h = \frac{V}{\pi r^2}$$

---

2. Linear Inequalities & Compound Intervals

Solving single-variable linear inequalities mirrors equation solving with one mandatory rule:

The Sign Inversion Rule: Whenever you multiply or divide both sides of an inequality by a negative number, you MUST reverse (flip) the direction of the inequality sign ($<$ becomes $>$, $\le$ becomes $\ge$, and vice versa).

Worked Example:   -4x + 9 ≤ 29
                  -4x ≤ 20
                  Divide by -4 (FLIP SIGN!):  x ≥ -5

Number Line Graphing Conventions:

• $<$ or $>$ (Strict Inequalities): Indicated by an Open Circle $(\circ)$ at the boundary point.
• $\le$ or $\ge$ (Inclusive Inequalities): Indicated by a Closed/Solid Circle $(\bullet)$ at the boundary point.

Compound Inequalities: "AND" vs. "OR"

1. Conjunction ("AND" / Bounded Double Inequality): Solution is the overlapping intersection of both conditions. $$-5 \le 3x + 1 < 13 \implies -6 \le 3x < 12 \implies -2 \le x < 4$$ (Graph: A continuous line segment between closed circle at $-2$ and open circle at $4$.)
2. Disjunction ("OR" / Disjoint Rays): Solution is the union of two separate conditions pointing in opposite directions. $$2x - 7 < -11 \quad \text{or} \quad 4x + 3 \ge 19$$ $$2x < -4 \implies x < -2 \quad \text{or} \quad 4x \ge 16 \implies x \ge 4$$

---

3. Absolute Value Equations & Inequalities

Geometrically, $|u|$ measures distance from zero on the number line. Because distance is non-negative, equations and inequalities split into specific cases:

1. Absolute Value Equations: $|ax + b| = c$

• If $c < 0$, there is no solution ($|u| = -5$ is impossible).
• If $c \ge 0$, split into two linear equations: $ax + b = c \quad \text{or} \quad ax + b = -c$.
• Example: $|2x - 3| = 9 \implies 2x - 3 = 9 \implies 2x = 12 \implies x = 6$, or $2x - 3 = -9 \implies 2x = -6 \implies x = -3$.

2. Absolute Value Inequalities: "Less ThAND" vs. "GreatOR"

+-----------------------------------------------------------------------------------------+
|                            ABSOLUTE VALUE INEQUALITY RULES                              |
+------------------------------------+----------------------------------------------------+
| LESS THAN: |u| ≤ c ("Less thAND")   | Bounded Conjunction:  -c ≤ u ≤ c                  |
| Example: |3x - 5| ≤ 16             | -16 ≤ 3x - 5 ≤ 16 ⇒ -11 ≤ 3x ≤ 21 ⇒ -11/3 ≤ x ≤ 7  |
+------------------------------------+----------------------------------------------------+
| GREATER THAN: |u| ≥ c ("GreatOR")  | Disjoint Union:  u ≥ c  OR  u ≤ -c                 |
| Example: |2x + 7| > 13             | 2x + 7 > 13 ⇒ 2x > 6 ⇒ x > 3                    |
|                                    | OR: 2x + 7 < -13 ⇒ 2x < -20 ⇒ x < -10             |
+------------------------------------+----------------------------------------------------+

---

4. Systems of Two Linear Equations ($2 \times 2$)

A simultaneous system of two linear equations represents two straight lines in the Cartesian coordinate plane. The geometric relationship determines the solution:

Method 1: Substitution (Best when a variable coefficient is $\pm 1$)

$$\begin{cases} x - 3y = 4 \\ 2x + 5y = 19 \end{cases}$$

1. Isolate $x$: $x = 3y + 4$
2. Substitute into equation 2: $2(3y + 4) + 5y = 19 \implies 6y + 8 + 5y = 19 \implies 11y = 11 \implies y = 1$
3. Back-substitute: $x = 3(1) + 4 = 7$. Solution: $(7, 1)$.

Method 2: Elimination / Addition (Best when coefficients can be aligned)

$$\begin{cases} 3x + 4y = 7 \\ 5x - 2y = 29 \end{cases}$$

1. Multiply equation 2 by 2 to align $y$-coefficients: $$2(5x - 2y = 29) \implies 10x - 4y = 58$$
2. Add to equation 1 to eliminate $y$: $$(3x + 10x) + (4y - 4y) = 7 + 58 \implies 13x = 65 \implies x = 5$$
3. Substitute $x = 5$ into equation 1: $$3(5) + 4y = 7 \implies 15 + 4y = 7 \implies 4y = -8 \implies y = -2$$
4. Solution: $(5, -2)$.

---

5. Solving Quadratic Equations & The Quadratic Formula

A quadratic equation has standard form $ax^2 + bx + c = 0$ (where $a \neq 0$).

Method 1: Factoring and the Zero Product Property

If $A \cdot B = 0$, then $A = 0$ or $B = 0$.

$$2x^2 - 7x + 3 = 0 \implies (2x - 1)(x - 3) = 0$$ $$2x - 1 = 0 \implies x = \frac{1}{2}, \quad x - 3 = 0 \implies x = 3$$

Method 2: The Quadratic Formula

When a quadratic expression cannot be factored by inspection, use the Quadratic Formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Discriminant Analysis ($D = b^2 - 4ac$)

The value under the radical ($D$) dictates the number and algebraic type of solutions:

                       DISCRIMINANT: D = b² - 4ac
                                   |
          +------------------------+------------------------+
          |                                                 |
        D > 0                                             D = 0                  D < 0
  Two Distinct Real Roots                          One Repeated Real Root    No Real Roots
(Rational if D is perfect square,                  x = -b / (2a)          (Two Complex Conjugate
 Irrational if not perfect square)                                         Roots: x = u ± vi)

Step-by-Step Worked Quadratic Example:

$$\text{Solve: } 3x^2 - 5x - 2 = 0$$

• Identify coefficients: $a = 3, b = -5, c = -2$
• Calculate discriminant: $D = (-5)^2 - 4(3)(-2) = 25 + 24 = 49$ (Positive perfect square $\rightarrow$ two rational roots)
• Apply formula: $$x = \frac{-(-5) \pm \sqrt{49}}{2(3)} = \frac{5 \pm 7}{6}$$ $$x_1 = \frac{5 + 7}{6} = \frac{12}{6} = 2, \quad x_2 = \frac{5 - 7}{6} = \frac{-2}{6} = -\frac{1}{3}$$