Let's solve each inequality separately:
Expand the expression42x - 84x^2 + 56 - 112x
Rearrange the terms to simplify-84x^2 - 70x + 56
Now we need to find the critical points by setting the expression equal to 0-84x^2 - 70x + 56 = 0
Now we solve for x using the quadratic formulax = (-(-70) ± √((-70)^2 - 4(-84)56))/(2*(-84))
x = (70 ± √(4900 + 18816))/(-168)
x = (70 ± √23716)/(-168)
Now we have the two possible critical points for x.
Now we check the intervals determined by the critical points and the endpoints of the interval, which are (3x+4)(14-28x) = 0 or undefined.
Expand the expression54x - 270 - 162x + 810
Simplify-162x + 54x - 270 + 810
Now we need to find the critical points by setting the expression equal to 0-108x + 540 = 0
Now solve for xx = 540/10x = 5
Now we check the intervals determined by the critical points and the endpoints of the interval, where the expression is equal to 0 or undefined.
I hope this helps! Let me know if you need further clarification.
Let's solve each inequality separately:
(3x+4)(14-28x) > 0Expand the expression
42x - 84x^2 + 56 - 112x
Rearrange the terms to simplify
-84x^2 - 70x + 56
Now we need to find the critical points by setting the expression equal to 0
-84x^2 - 70x + 56 = 0
Now we solve for x using the quadratic formula
x = (-(-70) ± √((-70)^2 - 4(-84)56))/(2*(-84))
x = (70 ± √(4900 + 18816))/(-168)
x = (70 ± √23716)/(-168)
Now we have the two possible critical points for x.
Now we check the intervals determined by the critical points and the endpoints of the interval, which are (3x+4)(14-28x) = 0 or undefined.
(9x-27)(6x-30) > 0Expand the expression
54x - 270 - 162x + 810
Simplify
-162x + 54x - 270 + 810
Now we need to find the critical points by setting the expression equal to 0
-108x + 540 = 0
Now solve for x
x = 540/10
x = 5
Now we check the intervals determined by the critical points and the endpoints of the interval, where the expression is equal to 0 or undefined.
I hope this helps! Let me know if you need further clarification.