4.4 Absolute Value Equations & Inequalities in One and Two Variables

4.4 Absolute Value Equations & Inequalities in One and Two Variables

Absolute value relations formalize distance metrics on the real number line and in coordinate space. On the FTCE Mathematics 6–12 examination, secondary educators must possess complete command of piecewise absolute value definitions, algebraic branching algorithms, extraneous solution detection, and two-variable transformations on the Cartesian plane.

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1. Geometric Foundation of Absolute Value as Metric Distance

The absolute value of a real number $x \in \mathbb{R}$, denoted $|x|$, is formally defined piecewise as:

$$|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$$

Equivalently, it can be expressed via the principal radical function as $|x| = \sqrt{x^2}$.

Geometrically, absolute value represents the Euclidean metric (distance) on the real number line. The quantity $|x|$ is the non-directed distance between the real number $x$ and the origin $0$. More generally, the expression $|x - c|$ represents the distance between the variable $x$ and a fixed center coordinate $c$:

$$d(x, c) = |x - c|$$

Because distance is fundamentally non-negative, the absolute value function satisfies the following core metric axioms:

1. Non-negativity: $|x| \ge 0$ for all $x \in \mathbb{R}$, with $|x| = 0 \iff x = 0$.
2. Symmetry: $|-x| = |x|$ and $|a - b| = |b - a|$.
3. Multiplicativity: $|ab| = |a||b|$ and $\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$ (for $b \neq 0$).
4. Triangle Inequality: $|a + b| \le |a| + |b|$ for all $a, b \in \mathbb{R}$.

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2. Solving Absolute Value Equations & Extraneous Branch Analysis

When solving an absolute value equation in one variable, the expression inside the absolute value bars must first be isolated. In standard isolated form $|ax + b| = c$:

• Case 1 ($c > 0$): Yields two distinct linear branches: $$ax + b = c \quad \text{or} \quad ax + b = -c$$
• Case 2 ($c = 0$): Yields exactly one linear equation $ax + b = 0 \implies x = -\frac{b}{a}$.
• Case 3 ($c < 0$): Has no real solution ($\emptyset$), because $|ax + b| \ge 0$ cannot equal a strictly negative quantity.

When the right-hand side contains variable terms, such as $|ax + b| = cx + d$, students must establish the domain restriction $cx + d \ge 0$ prior to branching, or systematically test all candidate solutions in the original equation. Roots that satisfy the unconstrained algebraic branches but cause $cx + d < 0$ are extraneous solutions created by the fact that $|u| = v \implies u^2 = v^2$, which also introduces roots to $u = -v$ when $v < 0$.

For equations with absolute values on both sides, $|ax + b| = |cx + d|$, the expression is resolved by considering $ax + b = \pm(cx + d)$.

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3. Absolute Value Inequalities: Conjunction vs. Disjunction

The algebraic structure of absolute value inequalities depends fundamentally on the direction of the inequality symbol relative to the isolated absolute value:

• Special Cases with Non-Positive Constants:
  • $|ax + b| < -k$ (where $k > 0$): No solution ($\emptyset$), since non-negative values are never strictly less than a negative number.
  • $|ax + b| \ge -k$ (where $k > 0$): Solution set is all real numbers $\mathbb{R} = (-\infty, \infty)$.
  • $|ax + b| \le 0$: Equivalent to the single point equation $ax + b = 0$.

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4. Coordinate Graphing of Absolute Value Relations & Transformations

In the two-dimensional Cartesian plane, the absolute value function $f(x) = |x|$ forms a characteristic V-shaped graph with vertex at $(0, 0)$ and two linear rays of slope $+1$ (for $x \ge 0$) and $-1$ (for $x < 0$).

The general vertex form is:

$$y = a|x - h| + k$$

• Vertex: The sharp turning point occurs at $(h, k)$.
• Axis of Symmetry: The vertical line $x = h$ bisects the figure into symmetric halves.
• Vertical Stretch/Reflection ($a$): If $a > 0$, the V opens upward with right ray slope $+a$ and left ray slope $-a$. If $a < 0$, the graph reflects across the horizontal axis, opening downward with vertex at $(h, k)$.
• Two-Variable Inequalities:
  • $y \le a|x - h| + k$ (with $a > 0$): Solid V-shaped boundary, shaded below the V (the region containing points with lesser $y$-values).
  • $y > a|x - h| + k$ (with $a > 0$): Dashed V-shaped boundary, shaded strictly above / inside the interior of the V.

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5. Worked Exemplar: Multi-Step Inequality with Sign Reversals

Problem: Solve the inequality $-4|2x - 3| + 7 \le -17$ completely over the set of real numbers. Express the solution set in set-builder and interval notation.

Step 1: Isolate the absolute value expression Subtract $7$ from both sides:

$$-4|2x - 3| \le -17 - 7 \implies -4|2x - 3| \le -24$$

Divide both sides by $-4$. Because we are dividing by a negative number, the inequality symbol must reverse:

$$|2x - 3| \ge \frac{-24}{-4} \implies |2x - 3| \ge 6$$

Step 2: Classify the compound structure The inequality is in "greater than or equal to" form ($|u| \ge c$ with $c = 6 > 0$). This constitutes a disjunction ("OR") representing points whose distance from the center is at least $6$:

$$2x - 3 \ge 6 \quad \text{or} \quad 2x - 3 \le -6$$

Step 3: Solve each linear branch independently

• Branch 1: $$2x - 3 \ge 6 \implies 2x \ge 9 \implies x \ge \frac{9}{2}$$
• Branch 2: $$2x - 3 \le -6 \implies 2x \le -3 \implies x \le -\frac{3}{2}$$

Step 4: Formulate the final solution representations

• Set-Builder Notation: $$\left\{ x \in \mathbb{R} \;\middle|\; x \le -\frac{3}{2} \;\text{ or }\; x \ge \frac{9}{2} \right\}$$
• Interval Notation: $$\left( -\infty, -\frac{3}{2} \right] \cup \left[ \frac{9}{2}, \infty \right)$$

Step 5: Verify using test values

• Choose $x = 0$ (in the excluded region $\left(-\frac{3}{2}, \frac{9}{2}\right)$): $$-4|2(0) - 3| + 7 = -4(3) + 7 = -12 + 7 = -5 \not\le -17 \quad (\text{False, correctly excluded})$$
• Choose $x = 5$ (in the right branch $\left[\frac{9}{2}, \infty\right)$): $$-4|2(5) - 3| + 7 = -4(7) + 7 = -28 + 7 = -21 \le -17 \quad (\text{True, correctly included})$$