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N is a negative real number. Which of the following is not true?
The absolute value of any number N is always nonnegative.
N = N, when N > 0 and N = N, when N < 0.
Since N is a negative number, therefore N = N is not true.
The number of solutions of the equation x^{2}  4 = 4 + x^{2} is
x^{2}  4 = 4 + x^{2}
Case I: If x^{2}  4 ≥ 0
i.e. x ≤ 2 and x ≥ 2
Then, x^{2}  4 = x^{2}  4
⇒ The given equation becomes x^{2}  4 = 4 + x^{2}.
⇒ 4 = 4; which is not possible.
Case II: If x^{2}  4 < 0
⇒ 2 < x < 2
Then, x^{2}  4 =  (x^{2}  4)
Thus, the given equation becomes  (x^{2}  4) = 4 + x^{2}.
Or 2x^{2} = 0
⇒ x = 0
Hence, the given equation has only one solution.
Take x = 2, then option (1), (2) will become true and hence those are not the answers.
Take x = – 2, option (4) will be true and hence not the answer.
But for any value of x, 2x > x.
So option (3) is not true at all. It is always false.
A, B, C and D are four friends and they are having w, x, y and z amount, respectively, such that w  z > 0, x  z < 0 and y  z < 0. Which of the following is necessarily true?
w  z > 0
x  z < 0; this means z  x > 0
y  z < 0; this means z  y > 0
The product of cubes of three positive numbers will also be positive.
If a < b, then the solution of x^{2}  (a + b) x + ab < 0 is
x^{2}  (a + b) x + ab < 0
(x  a) (x  b) < 0
a < x < b
What values of 'm' satisfy the inequality 3m^{2}  21m + 30 < 0?
3m^{2}  21m + 30 < 0
m^{2}  7m + 10 < 0
⇒ (m  5)(m  2) < 0
⇒ 2 < m < 5
What is the best description of 'x' which satisfies the inequality x^{2}  5x + 6 ≤ 0 ?
x^{2}  5x + 6 ≤ 0
⇒ (x  3)(x  2) ≤ 0
⇒ x ∈ [2, 3] or 2 ≤ x ≤ 3
Therefore, option 4 is the correct answer.
x is always positive.
If x is positive, x – x = 0.
If x is negative, x – x < 0.
Solve the system of inequalities:
5x + 2 > 3x  1
3x + 1 > 7x  4
(i) 5x + 2 > 3x  1 or 2x > 3
(ii) 3x + 1 > 7x  4
At x = 0, inequality is not satisfied. Thus, option (c) is rejected. Also x = 0 is not a solution of the equation. Since, this is a continuous function, the solution cannot start from 0. Thus options (a) and (b) are not right. Further, we see that the given function is quadratic with real roots. Hence, option (d) is also rejected.
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