Solve the inequality -2x^2 + 8 <= 0.

Step-by-step solution

1. 1

   Isolate the squared term. Subtract $8$ from both sides:

   $-2x^{2}\le-8$

2. 2

   Divide by $-2$ and reverse the inequality. Dividing (or multiplying) an inequality by a negative number flips its direction — the single rule that governs this problem:

   $x^{2}\ge4$

   Keeping $\le$ here would give the exactly wrong answer $-2\le x\le2$, so the flip is not optional.

3. 3

   Solve $x^{2}\ge4$ as a distance statement. Since $x^{2}=|x|^{2}$, the condition is $|x|\ge2$: the point $x$ is at least $2$ away from the origin. That is satisfied on the two rays outside $[-2,2]$, not between them:

   $x\le-2\qquad\text{or}\qquad x\ge2$

4. 4

   Write the solution set.

   $(-\infty,-2]\cup[2,\infty)$

   Both endpoints are included: at $x=\pm2$, $-2(4)+8=0$, and $0\le0$ is true ✓.

5. 5

   Confirm with the graph and by testing. $y=-2x^{2}+8$ is a downward parabola with roots at $\pm2$ and vertex $(0,8)$; it is at or below zero exactly outside the roots. Testing: $x=0$ gives $8\le0$ ✗ (correctly excluded), $x=3$ gives $-18+8=-10\le0$ ✓, $x=-3$ gives $-10\le0$ ✓. The raw inequality and the claimed set agree at $51$ exact rational points ✓.