Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-

Solution

• An inequality, \((x+1) (12-3x)^5 (x+3) (5-x) < 0\)

• The range of x, that satisfies the above inequality

• So, if we observe carefully, the inequality given to us can be written as,

o \((x+1) (12-3x) (12-3x)^4 (x+3) (5-x) < 0\)
• We know that, any number of the form \(N^2\) is always positive, expect for N = 0.

o So, we can say that, \((12-3x)^4\) is always positive, expect for x = 4,
o Thus, the inequality can be written as (x+1) (12-3x) (x+3) (5-x) < 0
• Now, let’s multiply the inequality by -1, twice, to make the coefficients of x, in 12-3x and 5-x, positive

o And, note that the inequality sign does not change as we are multiplying it by -1, twice
• Thus, the inequality becomes, (x+1) (3x - 12) (x+3) (x - 5) < 0
• The zero points of this inequality are x = -1, x = 4, x = -3 and x = 5
• Plot these points on the number line.

o Since, 5 is the greatest among all, the inequality will be positive, for all the points to the right of 5, on the number line
o And, it is negative, in the region, between 4 and 5
o It is again positive in the region, between -1 and 4
o And, it is again negative in the region, between -3 and -1
o And, it is positive for all the values of x, less than -3

Solution

• An inequality, \((x+1) (12-3x)^5 (x+3) (5-x) < 0\)

• The range of x, that satisfies the above inequality

• So, if we observe carefully, the inequality given to us can be written as,

o \((x+1) (12-3x) (12-3x)^4 (x+3) (5-x) < 0\)
• We know that, any number of the form \(N^2\) is always positive, expect for N = 0.

o So, we can say that, \((12-3x)^4\) is always positive, expect for x = 4,
o Thus, the inequality can be written as (x+1) (12-3x) (x+3) (5-x) < 0
• Now, let’s multiply the inequality by -1, twice, to make the coefficients of x, in 12-3x and 5-x, positive

o And, note that the inequality sign does not change as we are multiplying it by -1, twice
• Thus, the inequality becomes, (x+1) (3x - 12) (x+3) (x - 5) < 0
• The zero points of this inequality are x = -1, x = 4, x = -3 and x = 5
• Plot these points on the number line.

o Since, 5 is the greatest among all, the inequality will be positive, for all the points to the right of 5, on the number line
o And, it is negative, in the region, between 4 and 5
o It is again positive in the region, between -1 and 4
o And, it is again negative in the region, between -3 and -1
o And, it is positive for all the values of x, less than -3