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

Let's start by expanding the left side of the inequality using the distributive property:

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

Simplifying this expression, we get:

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

Combining like terms, we get:

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

Subtracting 35x^2 and 14x from both sides, we get:

x^2 - x - 13 > 0

Using the quadratic formula, we can find the roots of this quadratic equation:

x = (1 ± sqrt(1 - 4(1)(-13))) / 2 x = (1 ± sqrt(53)) / 2

Since the coefficient of the x^2 term is positive, the parabola opens upwards and the inequality is true for values of x that are outside the roots of the equation. Therefore, the solution set is:

x < (1 - sqrt(53)) / 2 or x > (1 + sqrt(53)) / 2

Alternatively, we can use a sign chart to determine the solution set:

diffCopy code
   +------------+------------+------------+
   |   x^2 - x   |   x^2 - x   |   x^2 - x   |
   |     < 0    |     = 0    |     > 0    |
---+------------+------------+------------+
 x |  x < 0 or   |   x = 1    |  x < 1 or  |
   |  x > 1     |            |  x > 0     |
---+------------+------------+------------+
+x |  x < -13/12 |            |   x > 13/12|
---+------------+------------+------------+

The sign chart shows that the inequality is true for values of x less than -13/12 or greater than 13/12, which is equivalent to the solution set we found using the quadratic formula.