How many integer values of x satisfy the equation |x-3|+|x-5|

Question

How many integer values of x satisfy the equation |x-3|+|x-5|<|x-11|?

Suraj Pandey · Accepted Answer

Understand the problemWe need to find the number of integer solutions for the inequality |x-3| + |x-5| < |x-11|.
Consider the critical points 3, 5, and 11Break the inequality into cases based on these points.
Case 1: x < 3|x-3| = 3-x, |x-5| = 5-x, |x-11| = 11-x3-x + 5-x < 11-x8-2x < 11-x8-2x < 11-x-2x + x < 11-8-x < 3x > -3
Case 2: 3 &le; x < 5|x-3| = x-3, |x-5| = 5-x, |x-11| = 11-xx-3 + 5-x < 11-x2 < 11-xx < 9
Case 3: 5 &le; x < 11|x-3| = x-3, |x-5| = x-5, |x-11| = 11-xx-3 + x-5 < 11-x2x-8 < 11-x3x < 19x < 19/3x < 6.33
Case 4: x &ge; 11|x-3| = x-3, |x-5| = x-5, |x-11| = x-11x-3 + x-5 < x-112x-8 < x-11x < -3 (No solutions)
Combine the casesFrom cases 1 and 2:-3 < x < 9
From case 3:5 &le; x < 6.33
Find integer valuesCombining the valid ranges:5 &le; x < 9
Integer values: 5, 6, 7, 8
ConclusionThere are 4 integer values of x that satisfy the inequality.