Solve the inequality -7 > 3x(x - 2).

Step-by-step solution

1. 1

   Expand the right-hand side.

   $3x(x-2)=3x^2-6x\quad\Longrightarrow\quad -7>3x^2-6x$

2. 2

   Rearrange into standard form. Bring everything to the side that keeps $x^2$ positive; subtracting $-7$ and reversing the reading gives

   $3x^2-6x+7<0$

   (Reading $-7>E$ as $E<-7$ is the same statement, just written with the expression first.)

3. 3

   Compute the discriminant. With $a=3$, $b=-6$, $c=7$:

   $\Delta=(-6)^2-4(3)(7)=36-84=-48<0$

   No real roots, so no sign changes anywhere.

4. 4

   Identify the constant sign. Because $a=3>0$, the parabola opens upward and stays above the axis. Completing the square quantifies it:

   $3x^2-6x+7=3(x-1)^2+4\ge 4$

5. 5

   Conclude and check. A quantity that is always at least $4$ is never negative, so the inequality has no solution. Spot-check the vertex $x=1$: $3(1)(1-2)=-3$, and $-7>-3$ is false $\checkmark$ — even at the most favourable point the inequality fails.