If x is an integer and it satisfies the inequalities, x^2 – 6x - 7...

Solution

• We are given that x is an integer, and
• We are also given two inequalities,

o \(x^2 – 6x - 7 < 0\)
o \(2x – x^2 + 3 < 0\)

• We need to find out which among the given answer choices can be a possible value of x that satisfies the given information

• We need to solve both the quadratic inequalities to arrive at the answer.
• Let’s solve the first inequality, \(x^2 - 6x – 7 < 0\)

o Implies, (x - 7)(x + 1) < 0

• Thus, the expression is negative for -1 < x < 7
• The values of x that satisfy this inequality are, x = {0, 1, 2, 3, 4, 5, 6}

o Multiplying by -1, we get, \(x^2 – 2x - 3 > 0\)
o Implies, (x - 3)(x + 1) > 0

• Thus, the expression is positive for x < -1 and x > 3

• Since, x is an integer, x = {4 , 5 , 6}

• Thus, the expression is negative for -1 < x < 7

• Thus, the expression is positive for x < -1 or x > 3

• Since, x is an integer, the values of x that satisfy the given expression are {4, 5, 6}