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📐 Mathematics  ·  Linear Inequalities  ·  JEE

A shopkeeper wants the cost price x of an item (in rupees) to satisfy: at least Rs 20 and the selling price (x + 0.25x) to not exceed Rs 100. What is the valid range for x?

Answer: 20 <= x <= 80.

  • A 20 <= x <= 80
  • B 20 <= x <= 100
  • C 0 <= x <= 80
  • D 20 <= x <= 75

Correct answer: A. 20 <= x <= 80

Explanation: Two conditions: (1) x ≥ 20. (2) Selling price = x+0.25x = 1.25x ≤ 100, so x ≤ 100/1.25 = 80. Intersection of both: 20 ≤ x ≤ 80. Answer: 20 ≤ x ≤ 80

Number Line vs Half-Plane Representationx > 2 (open circle, hollow)2x ≤ 2 (closed circle, filled)shaded regionsatisfies the inequality2D inequality: dashed line = strict, shaded = solution side

One-variable inequalities use open (excluded) or closed (included) circles on a number line; two-variable inequalities use a dashed (strict) or solid (non-strict) boundary line, with the solution being the entire shaded half-plane on one side of it.

Concept context

Solving linear inequalities in one and two variables algebraically and graphically, and representing solutions on a number line or as a region in a plane.

Read the full Linear Inequalities notes →