Question

(6x-1)(1+6x)+14<7x(2+5x)

Answer 1

6x+36x^2-1-6x+14<14x+35x^2


36x^2-35x^2-14x+13<0


x^2-14x+13<0


D=196-52=144


x1=(14+12)/2*1=26/2=13

x2=(14-12)/2*1=2/2=1


(1;13)

Answer 2

To solve the inequality, let's simplify both sides of the equation first:

Starting with the left-hand side (LHS):

LHS = (6x - 1)(1 + 6x) + 14 = 6x(1 + 6x) - 1(1 + 6x) + 14 = 6x + 36x^2 - 1 - 6x + 14 = 36x^2 + 5x + 13

Now, let's simplify the right-hand side (RHS):

RHS = 7x(2 + 5x) = 14x + 35x^2

The inequality becomes:

36x^2 + 5x + 13 < 14x + 35x^2

Rearranging the terms, we have:

0 < 35x^2 + 5x - 14x - 13

Combining like terms:

0 < 35x^2 - 9x - 13

Now, let's solve the quadratic inequality by factoring or using the quadratic formula. However, after calculating the discriminant (b^2 - 4ac), we find that it is negative, indicating that there are no real solutions. Therefore, we cannot solve this inequality analytically.

Graphically, we can plot the functions y = 35x^2 - 9x - 13 and y = 0 to determine the regions where the inequality holds true. The area below the curve of y = 35x^2 - 9x - 13 represents the values of x for which the inequality is satisfied.

Please note that due to the nature of text-based communication, I'm unable to provide an accurate graphical representation here. It's recommended to use a graphing calculator or software to visualize the graph and determine the solution intervals.