絶対値不等式計算機｜解集合と区間表記

An absolute value inequality is a mathematical statement that uses an inequality sign (<, >, ≤, or ≥) together with an absolute value expression. The absolute value |ax + b| measures the distance of the expression ax + b from zero on the number line. Asking when that distance satisfies an inequality condition defines a set of values for x, known as the solution set or solution interval. This calculator handles inequalities of the standard form |ax + b| ○ c, where a is the non-zero coefficient of x, b is the constant inside the absolute value, c is the constant on the right side, and ○ is one of the four inequality operators. The structure of the solution depends on which operator you use and on the relationship between c and zero. For less-than inequalities (< or ≤): |ax + b| < c is equivalent to −c < ax + b < c, which produces a bounded solution interval. The solution set is the range of x values where ax + b stays within distance c of zero. When c ≤ 0, a strict less-than inequality has no solution because absolute values cannot be less than a non-positive number. For greater-than inequalities (> or ≥): |ax + b| > c is equivalent to ax + b > c or ax + b < −c, producing two separate unbounded rays. The solution set is the union of all x values where ax + b is at least c units away from zero in either direction. When c < 0, every real number satisfies the inequality because absolute values are always ≥ 0 > c. When c = 0 and the operator is >, the solution is all real numbers except the single point x = −b/a. Absolute value inequalities model tolerance and range requirements in countless real-world settings. In manufacturing, the acceptable range of a machine part dimension is expressed as |measurement − target| ≤ tolerance. In science, confidence intervals express the uncertainty of a measured value as |measured − true| ≤ error. Speed limits use absolute value logic: if you must stay within 5 mph of 60 mph, then |speed − 60| ≤ 5. Financial portfolios set acceptable risk bands using the same structure. The most common mistake when solving these inequalities by hand is applying the wrong rule for the two cases. Remember: for less-than, you get a single interval (compound inequality with AND); for greater-than, you get a union of two rays (compound inequality with OR). Never write a less-than solution as two separate rays — that represents a completely different set. Always pay attention to whether the operator is strict (<, >) or non-strict (≤, ≥) when writing the endpoints of your solution.