In short
Drag the slider for k in the equation |x - 2| = k and three completely different things happen depending on the sign of k:
- k > 0 → two solutions, one to the left of 2 and one to the right, sitting symmetrically at x = 2 \pm k.
- k = 0 → one solution, x = 2 (the centre, the only point at distance 0 from itself).
- k < 0 → no solution, because an absolute value can never be negative.
Three regimes, one knob. This is called the trichotomy of |x - c| = k, and recognising it on sight is foundational for CBSE Class 11 algebra.
You have already seen — in Absolute Value — Equations — that |x - 2| = 5 has two solutions, that |x - 2| = 0 has one, and that |x - 2| = -3 has none. Three different equations, three different answers.
But what if those are not really three different equations at all? What if they are one equation with a knob on it, and turning the knob walks you through all three behaviours in a single continuous motion?
That is exactly what the slider below does. The equation stays |x - 2| = k the whole time. Only k moves. Watch the solution set on the number line — it splits, it shrinks to a single point, it disappears entirely — all from one drag.
The slider
Slide $k$ from $-3$ all the way to $7$. Watch how the green dots split apart, collide into a single yellow dot at $k = 0$, and then vanish into the empty set when $k$ goes negative.
Three things to notice as you drag:
- When k > 0, the two green dots are always symmetric around the centre x = 2. The further apart they sit, the larger k is. Why: the two solutions are x = 2 + k and x = 2 - k, so their midpoint is exactly 2 regardless of k.
- When you slowly bring k down to 0, the two dots glide toward each other and finally collide at x = 2. They do not disappear — they merge. Why: the equation |x - 2| = 0 asks "which point is at distance zero from 2?" The only such point is 2 itself, so the two solutions degenerate into one.
- The instant k drops below 0, both dots are gone — replaced by the empty set \varnothing. Why: the bars guarantee |x - 2| \ge 0 for every real x, so demanding it equal a negative number is asking the impossible.
The trichotomy strip
Here is the same story without the slider — three regimes side by side, each in its own colour.
The picture deserves a second look. Notice that the yellow strip in the middle is thin — it is a single value of k, not an interval. That is what makes k = 0 a boundary case: it is the exact moment where the two-solution regime collapses and the no-solution regime is about to begin. It is the topological hinge between the green and red worlds.
Three worked examples — one for each regime
Example 1 (k > 0): Solve |x − 2| = 5
This is the green regime. With k = 5 > 0, you expect two solutions.
Step 1. Split into cases. The expression x - 2 has absolute value 5, so x - 2 is either +5 or -5.
Why: |y| = 5 means y is exactly 5 units from zero, which is true for y = 5 and y = -5 — no other numbers.
Step 2. Solve each.
Step 3. Check on the number line. The number 7 is 5 units to the right of 2. The number -3 is 5 units to the left of 2. Both sit at distance 5 from the centre.
Result. x = 7 or x = -3. Two solutions. Set the slider to k = 5 and confirm the green dots land exactly there.
Example 2 (k = 0): Solve |x − 2| = 0
The yellow boundary case. The equation asks: which numbers are at distance zero from 2?
Step 1. Apply the definition. The absolute value of any expression is 0 if and only if that expression itself is 0.
Why: |y| = 0 forces y = 0 — non-negativity says |y| \ge 0 always, and the only way to achieve equality is at y = 0. There is no separate "negative case" to consider, because -0 = 0.
Step 2. Solve.
Step 3. Check. |2 - 2| = |0| = 0. Correct.
Result. x = 2. Exactly one solution. Set the slider to k = 0 and watch the two green dots collide into a single yellow dot at x = 2 — they do not disappear, they merge.
This is the only regime where the two cases of the case-split degenerate into the same equation. It is the algebraic shadow of the geometric fact that the two arrows of length zero from a point both end at that same point.
Example 3 (k < 0): Solve |x − 2| = −3
The red regime. No algebra is needed at all.
Step 1. Inspect. The left side is an absolute value, so it is \ge 0 for every real x. The right side is -3 < 0.
Why: a non-negative quantity can never equal a negative quantity. There is no real number that makes this work.
Step 2. Conclude.
Result. No solution. Set the slider anywhere with k < 0 — for instance k = -3 — and you will see the empty-set symbol \varnothing replace the dots entirely.
If you ever find yourself "solving" an equation like |2x + 5| = -7 on a CBSE Class 11 paper by writing two cases, stop. The answer is "no solution" the moment you read the equation. Splitting into cases here will give you two phoney "solutions" that fail the original equation, and you will lose marks for not noticing the obvious.
Why three cases and only three?
This trichotomy is not specific to the centre 2 or to the linear inside x - 2. It applies to any equation of the form |f(x)| = k:
- k > 0: the expression f(x) takes the value +k or -k, so you split into two equations f(x) = k and f(x) = -k, each potentially contributing solutions.
- k = 0: only f(x) = 0 is possible, so you solve a single equation.
- k < 0: impossible, no solution.
The reason this is a trichotomy — three cases and only three — comes from the trichotomy law for the real numbers: every real number k is either positive, zero, or negative, and these three states are mutually exclusive and exhaustive. Since the behaviour of |f(x)| = k is governed only by which of these three buckets k falls into, you get exactly three regimes — no more, no fewer.
This connection between sign-of-k and number-of-solutions is one of the cleanest examples in school algebra of how a parameter (something you can tune) controls the qualitative behaviour of an equation. As k slides through zero, the equation does not just change its answer — it changes its type. Mathematicians call this a bifurcation: a single value of a parameter at which the structure of the solution set abruptly reorganises itself. The same idea shows up later in calculus when you study how the number of solutions of f(x) = c depends on the height c, and again in physics when you study phase transitions.
In CBSE Class 11, this trichotomy turns up in nearly every absolute value question on the syllabus. Examiners love asking "for what values of k does the equation |x - a| = k have exactly two real solutions?" The answer — k > 0 — is one slider drag away from being obvious. They also like the contrapositive: "given that |3x - 7| = m has no solution, what can you say about m?" Answer: m < 0. And the boundary version: "for what value of m does |3x - 7| = m have exactly one solution?" Answer: m = 0, with that single solution being x = 7/3. All three are slices of the same trichotomy.
A useful mental check while solving any such problem: before reaching for case-splits, glance at the right-hand side. If it is positive, you are in the green regime and you will get two answers. If it is zero, you are at the yellow boundary and you will get one. If it is negative, you can put down your pen and write "no solution" — there is nothing to compute. That five-second check, drilled into reflex, will save you from an entire category of careless errors on exam day.
See also
- Absolute Value — Equations — the parent article with the full case-split method.
- The V-shaped graph: where |x − a| meets a horizontal line — the same trichotomy seen graphically: a horizontal line cuts the V in two, one, or zero places depending on its height.
- Absolute value equations as distance: arrows fanning out — the geometric "two arrows of length r from a" picture for the k > 0 regime.
References
- NCERT Mathematics, Class 11 — Chapter 6: Linear Inequalities (covers absolute value as distance, used implicitly throughout).
- Stewart, J., Redlin, L., Watson, S. — Precalculus: Mathematics for Calculus. The "absolute value equations" section motivates the three-case structure exactly as presented here.
- Wikipedia: Absolute value — definition, properties, and the metric interpretation of |x - y|.
- Wikipedia: Trichotomy (mathematics) — the formal statement of the law of trichotomy on the real numbers, which underwrites why exactly three regimes appear.