State which of the following statements is True or False
(i) If x < y and b < 0, then
(ii) If xy > 0, then x > 0 and y < 0
(iii) If xy > 0, then x < 0 and y < 0
(iv) If xy < 0, then x < 0 and y < 0
(v) If x < –5 and x < –2, then x ∈ (– ∞, – 5)
(vi) If x < –5 and x > 2, then x ∈ (– 5, 2)
(vii) If x > –2 and x < 9, then x ∈ (– 2, 9)
(viii) If |x| > 5, then x ∈ (– ∞, – 5) ∪ [5, ∞)
(ix) If |x| ≤ 4, then x ∈ [– 4, 4]
(x) Graph of x < 3 is
(xi) Graph of x ≥ 0 is
(xii) Graph of y ≤ 0 is
(xiii) Solution set of x ≥ 0 and y ≤ 0 is
(xiv) Solution set of x ≥ 0 and y ≤ 1 is
(xv) Solution set of x + y ≥ 0 is
(i) It is False.
x < y, b<0 ……. (given)
Multiplication or division by -ve no. Inverts the inequality sign
Thus, x/b > y/b
(ii) It is False.
If xy > 0, then,
Either x >0 & y > 0,
Or x < 0 & y < 0.
(iii) It is True.
If xy > 0, then,
Either x >0 & y > 0,
Or x < 0 & y < 0.
(iv) It is False.
If xy < 0, then,
Either x < 0 & y > 0,
Or x > 0 & y < 0
(v) It is True.
We know that,
x < -5 → x (-∞,-5) ………. (i)
& x < -2 → x (-∞,-2) ………. (ii)
Thus, x (-∞,-5) ………… [By taking intersection from (i) & (ii)]
(vi) It is False.
We know that,
x < -5 → x (-∞,-5) ………. (i)
& x < 2 → x (∞,2) ………. (ii)
Therefore, x has no common solution ……… [From (i) & (ii)]
(vii) It is True.
x > -2 → x (-∞,-2) ………. (i)
& x < 9 x (-∞,9) ………. (ii)
Therefore, x (-2,9) ……… [From (i) & (ii)]
(viii) It is True.
|x|< 5
Thus, there will be two cases,
x > 5 → x (5,∞) …… (i)
& -x > 5 → x < -5
→ x ( -∞,-2) …… (ii)
x (-∞,-5) U (5,∞) ……. [From (i) & (ii)]
(ix) It is True.
|x| ≤ 4,
Thus, there will be two cases,
x ≤ 4 → x (-∞,4) …….. (i)
& -x ≤4, x ≥ -4 → x [-4,∞] ……. (ii)
Therefore, x [-4,4] …….. [From (i) & (ii)]
(x) It is True.
Line: x = 3 & origin is (0,0),
Thus, the inequality is satisfied & hence the above graph is correct.
(xi) It is True.
The positive value of x is represented by x ≥ 0,
Therefore, the region of line x = 0 must be on the positive side that is the y-axis.
(xii) It is False.
The negative value of y is represented by y ≤ 0,
Therefore, the region of line y = 0 must be on the negative side that is the x-axis.
(xiii) It is False.
The shaded region is the first quadrant & the 4th quadrant is represented by x ≥ 0 & y ≤ 0.
(xiv) It is False.
The region on the left side of the y-axis is implied by x ≥ 0 & the region below the line y = 1 is implied by y ≤ 1.
(xv) It is True
The inequality is satisfied if we take any point above the line x + y = 0, say (3,2)
Thus, x+y ≥ 0
….…..(since, 3 + 2 = 5 > 0)
Therefore, the region should be above the line x+y = 0