Counting solutions satisfying modulus conditions

The question asks to count the number of values (integers or real numbers) satisfying a condition or equation involving absolute values, rather than finding them explicitly.

brazil-enem 2024 Q173 View
The number of integers satisfying $|x - 3| \leq 2$ is:
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
isi-entrance 2014 Q7 View
Find the number of solutions of $|2x - [x]| = 4$, where $[x]$ denotes the greatest integer function.
(A) 2 (B) 3 (C) 4 (D) Infinitely many
isi-entrance 2020 Q12 View
For a real number $x$, let $[ x ]$ denote the greatest integer less than or equal to $x$. Then the number of real solutions of $| 2 x - [ x ] | = 4$ is
(A) 4
(B) 3
(C) 2
(D) 1 .
jee-main 2021 Q61 View
The number of real solutions of the equation, $x ^ { 2 } - | x | - 12 = 0$ is:
(1) 2
(2) 3
(3) 1
(4) 4
jee-main 2021 Q72 View
The number of elements in the set $\{ x \in R : ( | x | - 3 ) | x + 4 | = 6 \}$ is equal to
(1) 3
(2) 2
(3) 4
(4) 1
jee-main 2023 Q61 View
The number of real roots of the equation $x | x | - 5 | x + 2 | + 6 = 0$, is
(1) 5
(2) 4
(3) 6
(4) 3
jee-main 2024 Q81 View
The number of distinct real roots of the equation $| x + 1 | | x + 3 | - 4 | x + 2 | + 5 = 0$, is
taiwan-gsat 2025 Q1 5 marks View
A point $P$ on a number line satisfies the condition that the distance from $P$ to 1 plus the distance from $P$ to 4 equals 4. How many such points $P$ are there?
(1) 0
(2) 1
(3) 2
(4) 3
(5) Infinitely many
turkey-yks 2015 Q10 View
What is the sum of all natural numbers such that when divided by 6, the quotient and remainder are equal to each other?
A) 84
B) 91
C) 96
D) 105
E) 112