निरपेक्ष मान समीकरण और असमानता सॉल्वर

निरपेक्ष मान (Absolute Value) क्या है?

The absolute value of a real number x, denoted |x|, measures its distance from zero on the number line — always a non-negative quantity. The formal definition is piecewise:

|x| = x if x ≥ 0     |x| = −x if x < 0

Equivalently, |x| = max(x, −x) = √(x²). The geometric interpretation is direct: |x| is the number of units between x and 0, so |5| = 5 and |−5| = 5. For expressions, |ax + b| measures the distance between ax + b and 0, making it central to tolerance, error analysis, and constraint problems.

|ax + b| = c (समीकरण) को हल करना

The key principle is that |expression| = c has solutions only when c ≥ 0. Split into two cases using the definition:

• Case 1 (positive branch): ax + b = c  →  x = (c − b) / a
• Case 2 (negative branch): ax + b = −c  →  x = (−c − b) / a

When a = 0 the equation becomes |b| = c, which is true for all x if |b| = c, and false for all x otherwise. Always verify each solution by substituting back.

|ax + b| < c (लघुतर असमानता) को हल करना

A less-than absolute value inequality means "the expression is within c units of zero." It is equivalent to the compound inequality:

−c < ax + b < c

Solve both parts simultaneously by isolating x. The solution is always a bounded open interval centred at x = −b/a with radius c/|a|. If c ≤ 0, there is no solution. The number line shows a shaded segment between two open endpoints.

|ax + b| > c (गुरुतर असमानता) को हल करना

A greater-than inequality means "the expression is more than c units from zero." Split into two separate inequalities:

ax + b > c   OR   ax + b < −c

The solution is a union of two rays: (−∞, x₁) ∪ (x₂, +∞). If c < 0, every real number is a solution since absolute value is always ≥ 0. The number line shows shading extending outward in both directions from two open points.

|ax + b| = |cx + d| (दोहरा निरपेक्ष मान) को हल करना

When two absolute values are set equal, use the fact that |A| = |B| means A = B or A = −B:

• Case 1: ax + b = cx + d  →  (a − c)x = d − b
• Case 2: ax + b = −(cx + d)  →  (a + c)x = −d − b

Each case yields at most one solution. Special handling is needed when a = c (Case 1 becomes trivial) or a = −c (Case 2 becomes trivial).

अंतराल संकेतन त्वरित संदर्भ (Interval Notation)

Inequality	Meaning	Interval Notation	Number Line
|x| = 3	x is 3 from 0	{-3, 3}	Two filled dots
|x| < 3	within 3 of 0	(-3, 3)	Open-interval shade
|x| ≤ 3	at most 3 from 0	[-3, 3]	Closed-interval shade
|x| > 3	more than 3 from 0	(-∞,-3) ∪ (3,+∞)	Two outward rays
|x| ≥ 3	at least 3 from 0	(-∞,-3] ∪ [3,+∞)	Two outward closed rays

Real-World Applications

• Engineering tolerances: A manufactured part is acceptable if |dimension − target| ≤ ε, directly an absolute value inequality.
• Error bounds: Measurement uncertainty is expressed as |measured − true| ≤ error margin.
• Temperature control: A thermostat keeps a room within ±2°C of the setpoint: |T − T₀| ≤ 2.
• Signal processing: Keeping a signal's deviation from a reference within a threshold.
• Quality control: Rejecting items whose weight differs from spec by more than a given amount: |w − w₀| > tol.
• Finance: Stop-loss rules: exit if |price − entry| > risk tolerance.