Absolute Value Calculator

The absolute value of a number x, written |x|, is its distance from zero on the number line, always a non-negative result. Formally: |x| = x if x ≥ 0, and |x| = −x if x < 0. So |5| = 5, |−5| = 5, and |0| = 0. The absolute value strips away the sign, leaving only the magnitude.

Split into two cases: 2x + 3 = 7 (giving x = 2) and 2x + 3 = −7 (giving x = −5). Always verify by substituting back: |2(2)+3| = |7| = 7 ✓ and |2(−5)+3| = |−7| = 7 ✓. If the right-hand side is negative (e.g. |2x+3| = −7), there is no solution since absolute value is always ≥ 0.

For |expression| < c: rewrite as −c < expression < c and solve the compound inequality. Here: −3 < x − 4 < 3, so 1 < x < 7, which is the interval (1, 7). For |expression| > c: split into expression > c or expression < −c, giving a union of two intervals. The key rule: < gives a bounded interval; > gives an unbounded union.

|x| < c means x is within distance c from zero: −c < x < c, written as the interval (−c, c). |x| > c means x is more than distance c from zero: x < −c or x > c, written as (−∞, −c) ∪ (c, +∞). The less-than case is bounded (a finite interval); the greater-than case is unbounded (two separate rays extending to infinity).

No - the absolute value |x| is always greater than or equal to zero for any real number x. This is why an equation like |x + 3| = −5 has no solution. It also means |x| ≥ 0 is an axiom of absolute value, and |x| = 0 if and only if x = 0.

On the number line, |x| is the distance from x to the origin (0). More generally, |x − a| is the distance from x to the point a. This geometric interpretation is why absolute value appears in distance formulas, error bounds, and tolerance specifications. For complex numbers, |z| = √(a² + b²) for z = a + bi, which is the distance from the origin in the complex plane.

Key properties: (1) |x| ≥ 0 for all x. (2) |x| = 0 ⟺ x = 0. (3) |−x| = |x| (symmetry). (4) |xy| = |x||y| (multiplicative). (5) |x/y| = |x|/|y| for y ≠ 0. (6) |x + y| ≤ |x| + |y| (triangle inequality - perhaps the most important property in analysis). (7) ||x| − |y|| ≤ |x − y| (reverse triangle inequality).

The triangle inequality states |x + y| ≤ |x| + |y| for all real numbers x and y. Geometrically: the length of one side of a triangle is at most the sum of the other two sides. In analysis, this inequality is fundamental - it appears in proofs of convergence, continuity, and the distance axioms of metric spaces. Equality holds when x and y have the same sign (or one is zero).

Absolute value quantifies deviation or distance without regard to direction: (1) Error and tolerance: a measurement is acceptable if |measured − target| ≤ 0.01. (2) Signal processing: the magnitude spectrum uses |z| for complex amplitudes. (3) Statistics: mean absolute deviation = average of |xᵢ − mean|. (4) Economics: percentage change uses |new − old| / old. (5) GPS and navigation: |lat₁ − lat₂| + |lon₁ − lon₂| is the Manhattan distance.

The graph of y = |x| is V-shaped: it follows y = x for x ≥ 0 and y = −x for x < 0, meeting at the vertex (0, 0). For y = |x − h| + k, the vertex shifts to (h, k). The slopes are +1 and −1 away from the vertex. For y = a|x − h| + k, the slopes become +a and −a; |a| > 1 steepens the V, |a| < 1 flattens it. The graph is always V-shaped (or U-shaped for even powers) and symmetric about the vertical line through the vertex.