Question

|2x-1|+|x-5|=7 find x.

Answer 1

To solve the given problem, break it into different cases,

Case 1) x ≤ 1/2

Here, the value of expression in the first modulus i.e., 2x - 1 is negative and thus it opens with a negative sign. The value of expression in the second modulus i.e., x - 5 is also negative and hence it opens with a negative sign.

Thus, -(2x-1) -(x-5) = 7

=> -2x + 1- x + 5 = 7

=> -3x + 6 = 7

=> x = -1/3

Case 2) 1/2 < x < 5

Here, the value of expression in the first modulus i.e., 2x - 1 is positive and thus it opens with a positive sign. The value of expression in the second modulus i.e., x - 5 is negative and hence it opens with a negative sign.

Thus, (2x-1) - (x-5) = 7

=> 2x - 1 - x + 5 = 7

=> x + 4 = 7

=> x = 3

Case 3) x ≥ 5

Here, the value of expression in the first modulus i.e., 2x - 1 is positive and thus it opens with a positive sign. The value of expression in the second modulus i.e., x - 5 is also positive and hence it opens with a positive sign.

Thus, 2x - 1 + x - 5 =7

=> 3x - 6 = 7

=> 3x = 13

=> x = 13/3

Hence, the values of x satisfing the equation |2x-1| + |x-5| = 7 are x = -1/3, 3 and 13/3.