Solve the following inequality.
(15(x^2 -1))>16x
Select the correct choice below and, if necessary, fill in the answer box.
A.The solution set is
B.There is no real solution
To solve the inequality (15(x^2 - 1)) > 16x, we first need to simplify it:
15x^2 - 15 > 16x
Rearrange the terms to bring all terms to one side:
15x^2 - 16x - 15 > 0
Now, we have a quadratic inequality. To solve it, we can find the roots of the corresponding quadratic equation:
15x^2 - 16x - 15 = 0
Solving this quadratic equation, we get:
x = (16 ± sqrt(16^2 - 4*15*(-15))) / (2*15)
x = (16 ± sqrt(256 + 900)) / 30
x = (16 ± sqrt(1156)) / 30
x = (16 ± 34) / 30
x = (16 + 34) / 30 or x = (16 - 34) / 30
x = 50/30 or x = -18/30
x = 5/3 or x = -3/5
Now, we need to test the inequality in the intervals created by these roots to determine the solution set. We'll test with values within each interval and see where the inequality holds true.
For x < -3/5:
Let's say x = -1
15(-1)^2 - 15 > 16(-1)
15(1) - 15 > -16
15 - 15 > -16
0 > -16
The inequality holds true for x < -3/5.
For -3/5 < x < 5/3:
Let's say x = 0
15(0)^2 - 15 > 16(0)
-15 > 0
The inequality does not hold true for -3/5 < x < 5/3.
For x > 5/3:
Let's say x = 2
15(2)^2 - 15 > 16(2)
15(4) - 15 > 32
60 - 15 > 32
45 > 32
The inequality holds true for x > 5/3.
Therefore, the solution set is x < -3/5 and x > 5/3.
A. The solution set is (-∞, -3/5) U (5/3, ∞)