An Absolute Value Equation With One Solution

An absolute value equation typically yields two solutions, a single solution, or no solution at all. While the standard form $|x| = a$ (where $a > 0$) produces two distinct answers ($x = a$ and $x = -a$), the scenario resulting in exactly one solution is a critical concept in algebra. It represents the precise moment the graph of the absolute value function touches the horizontal axis at its vertex. Understanding why and how this happens requires a solid grasp of the definition of absolute value, the structure of the equation, and the geometric interpretation of distance.

The Core Concept: Distance from Zero

At its heart, the absolute value of a number, denoted $|x|$, represents its distance from zero on the number line. Distance is inherently non-negative. That's why, the equation $|x| = a$ asks: "What numbers are exactly $a$ units away from zero?

• If $a > 0$, there are two numbers at that distance: $a$ and $-a$. This yields two solutions.
• If $a < 0$, no number exists at a negative distance. This yields no solution.
• If $a = 0$, there is exactly one number at a distance of zero from the origin: zero itself.

This third case is the foundation of an absolute value equation with one solution. On the flip side, the equation $|x| = 0$ simplifies directly to $x = 0$. There is no "plus or minus" scenario because $+0$ and $-0$ are identical.

The General Form: Vertex on the Axis

In algebra, we rarely encounter the bare equation $|x| = 0$. On the flip side, instead, we face expressions inside the absolute value bars, such as $|ax + b| = c$ or $|expression| = k$. For these equations to have a single solution, the vertex of the V-shaped graph must rest exactly on the x-axis.

Consider the standard form: $|ax + b| = c$

For a single solution to exist, the right-hand side ($c$) must be zero. $|ax + b| = 0$

Because the absolute value of any expression is zero only when the expression inside is zero, we drop the bars and solve the linear equation inside: $ax + b = 0$ $x = -\frac{b}{a} \quad (\text{provided } a \neq 0)$

This single value of $x$ is the only solution. It corresponds to the $x$-coordinate of the vertex of the function $y = |ax + b|$.

Example 1: Basic Linear Expression

Solve $|3x - 6| = 0$.

1. Set the inside expression equal to zero: $3x - 6 = 0$.
2. Solve for $x$: $3x = 6 \rightarrow x = 2$.
3. Solution set: ${2}$.

Verification: $|3(2) - 6| = |6 - 6| = |0| = 0$. The equation holds true only for $x=2$.

Equations with Non-Zero Constants: The "Impostor" Single Solution

A common point of confusion arises with equations like $|x - 5| = -3$. Because of that, a student might rush to set up two equations ($x-5=3$ and $x-5=-3$) and get two answers. That said, since an absolute value cannot equal a negative number, this equation has no solution.

Conversely, what about $|x - 5| = 3$? This has two solutions ($x=8, x=2$).

Is it possible to have a single solution when the right-hand side is not zero? Worth adding: the only exception occurs if the domain of $x$ is artificially restricted (e. No.g. If $|expression| = k$ and $k > 0$, the definition of absolute value demands two distinct inputs (opposites) that yield the same output $k$. , "Solve for $x > 0${content}quot;), but in the standard real number system $\mathbb{R}$, a positive constant on the right always yields two solutions or zero solutions (if the equation simplifies to a contradiction).

Because of this, the necessary and sufficient condition for a single unique solution in $\mathbb{R}$ is that the equation simplifies to $|expression| = 0$.

Advanced Scenarios: Quadratics and Nested Absolute Values

The principle remains the same even when the expression inside the bars is non-linear. That said, the equation $|f(x)| = 0$ is equivalent to $f(x) = 0$. The number of solutions depends entirely on the roots of $f(x)$ Not complicated — just consistent. Turns out it matters..

Quadratic Inside the Bars

Solve $|x^2 - 4x + 4| = 0$.

1. Recognize the condition: The inside must be zero.
2. $x^2 - 4x + 4 = 0$.
3. Factor: $(x - 2)^2 = 0$.
4. Solve: $x = 2$ (a double root).

Even though the quadratic has a "double root," the solution set for the absolute value equation is still just ${2}$. It is a single distinct solution Surprisingly effective..

Nested Absolute Values

Solve $||x - 1| - 2| = 0$.

1. Outer absolute value equals zero $\rightarrow$ Inner expression equals zero. $|x - 1| - 2 = 0$
2. $|x - 1| = 2$.
3. Now we have a standard absolute value equation with a positive constant on the right.
4. This yields two solutions: $x - 1 = 2 \rightarrow x = 3$; $x - 1 = -2 \rightarrow x = -1$.

Crucial Distinction: The original equation $||x - 1| - 2| = 0$ had a zero on the far right, but the inner equation resolved to a positive constant. This resulted in two solutions, not one. The "one solution" rule applies strictly to the outermost operation equaling zero and the resulting inner equation having only one root Worth keeping that in mind..

If the problem were $||x - 1| - 2| = k$:

• $k < 0$: No solution.
• $k = 0$: Solutions depend on roots of $|x-1|=2$ (Two solutions here).
• $k > 0$: Typically four solutions (two for each branch of the inner absolute value).

Geometric Interpretation: The Vertex Touch

Visualizing the graph $y = |ax + b|$ provides the most intuitive understanding.

• The graph is a V-shape.
• The bottom point of the V is the vertex.
• The vertex occurs where the expression inside is zero: $ax + b = 0$.
• The y-coordinate of the vertex is always 0.

When we solve $|ax + b| = c$, we are finding the intersection points of the V-shaped graph and the horizontal line $y = c$ It's one of those things that adds up. Practical, not theoretical..

• $c > 0$: The horizontal line cuts through the two arms of the V. Two intersections (Two solutions).
• $c < 0$: The horizontal line is below the x-axis. The V-shape sits on/above the x-axis. No intersections (No solution).
• $c = 0$: The horizontal line is the x-axis. It touches the V exactly at the vertex. One intersection (One solution).

This geometric view confirms algebraically why the right-hand side must be zero for a unique solution.

Solving Strategy: A Step-by-Step Checklist

When faced with an absolute value equation, follow this protocol to determine the solution count quickly:

1. Isolate the Absolute Value: Get the $|expression|$ alone on one side.
   • Example: $2|x - 3| + 4 = 4 \