Solve for x : (2x+3) (1-x) (x+5) ≥ 0
Solve for x : (2x+3) (1-x) (x+5) ≥ 0
The correct option is C
Given inequality can also be writtren as -(2x + 3)(x - 1)(x + 5) ≥ 0
⇒ (2x + 3)(x - 1)(x + 5) ≤ 0 (Multiplying with '-' sign changes the inequality)
Take the eqaution (2x + 3)(x - 1)(x + 5) = 0
⇒ x = −32, x =1, x = -5
Sign scheme of (2x + 3)(x - 1)(x + 5) is shown in the figure below.
The values of x for which (2x + 3)(x - 1)(x + 5) ≤ 0 is given from above sign - scheme as,
x ∈ (−∞,-5] U [−32,1]
Explanation of sign scheme:
In the above equation, the intervals that arise are
i) x ≥ 1 in which (2x + 3), (x+1) and (x+5) are all positive for any x. So,multiplication of all 3 factors is positive.
ii) −32 ≤ x ≤ 1: Here, (x-1) is negative and other 2 factors are positive. So, overall multiplication is negative.
iii) -5 ≤ x ≤ −32: Here, 2 factors (2x + 3), (x+1) are negative and the other factor is positive.Overall multiplication is hence positive.
iv) x ≤ -5 : All 3 factors are negative. So, overall multiplication is negative.
Given inequality can also be writtren as -(2x + 3)(x - 1)(x + 5) ≥ 0
⇒ (2x + 3)(x - 1)(x + 5) ≤ 0 (Multiplying with '-' sign changes the inequality)
Take the eqaution (2x + 3)(x - 1)(x + 5) = 0
⇒ x = −32, x =1, x = -5
Sign scheme of (2x + 3)(x - 1)(x + 5) is shown in the figure below.
The values of x for which (2x + 3)(x - 1)(x + 5) ≤ 0 is given from above sign - scheme as,
x ∈ (−∞,-5] U [−32,1]
Explanation of sign scheme:
In the above equation, the intervals that arise are
i) x ≥ 1 in which (2x + 3), (x+1) and (x+5) are all positive for any x. So,multiplication of all 3 factors is positive.
ii) −32 ≤ x ≤ 1: Here, (x-1) is negative and other 2 factors are positive. So, overall multiplication is negative.
iii) -5 ≤ x ≤ −32: Here, 2 factors (2x + 3), (x+1) are negative and the other factor is positive.Overall multiplication is hence positive.
iv) x ≤ -5 : All 3 factors are negative. So, overall multiplication is negative.