If x is an integer, then how many values of x will satisfy the equatio

Solution

• We are given that x is an integer, and
• We are also given an absolute value equation, |x + 2| = |3x + 14|

• We need to find the number of values of x, that satisfy the given equation

• In this question, we have modulus function on both sides of the equation.
• Let’s first apply the definition to the LHS of the equation.
• From this, we get,

o x + 2 = |3x + 14|, if x ≥ -2, and
o x + 2 = -|3x + 14|, if x < -2
• Now, let’s consider the first case, x + 2 = |3x + 14|, if x ≥ -2. And, if we remove the modulus sign on the RHS of this equation, we get,

o x + 2 = 3x + 14, if x ≥ -2 and x ≥ -14/3

 Solving this equation, we get, 2x = -12
 Implies, x = -6, which does not lie in the range
 Thus, x = -6 is not a possible value
o x + 2 = -(3x + 14), if x ≥ -2 and x < -14/3

 This case is not possible as no value of x can simultaneously be ≥ -2 and < -14/3
• Considering the second case, x + 2 = -|3x + 14|, if x < -2, and removing the modulus sign, we get,

o x + 2 = -(3x + 14), if x < -2 and x ≥ -14/3

 Solving this equation, we get, 4x = -16
 Implies, x = -4, which lies in the range [-14/3, -2)
 Thus, x = -4 is a possible value of x[/list]
o x + 2 = (3x + 14), if x < -2 and x < -14/3

 Solving this equation, we get, 2x = -12
 Implies, x = -6, which is less than both -2 and -14/3
 Thus, x = -6 is a possible value of x