f(x) >= d selects graph portions on or above y=d; f(x)>d excludes intersections. For <= or <, use portions on or below, or strictly below. Sketch the factored cubic, draw the comparison line and locate every intersection. Read intervals from left to right. Use the comparison intersections rather than the zeros of f, unless d=0. For f(x)=x(x-1)(x+1), f(x)=6 has just x=2, since f(x)-6=(x-2)((x+1)2+2). The curve is above the line exactly for x>2. Check: a cubic can meet a horizontal line once or three times. A single visible crossing on a cropped display is not proof that no others exist. Strict inequalities omit their boundary inputs.

\(f(x)\ge d\) selects graph portions on or above \(y=d\); \(f(x)>d\) excludes intersections. For \(\le\) or \(<\), use portions on or below, or strictly below.

Sketch the factored cubic, draw the comparison line and locate every intersection. Read intervals from left to right. Use the comparison intersections rather than the zeros of \(f\), unless \(d=0\).

For \(f(x)=x(x-1)(x+1)\), \(f(x)=6\) has just \(x=2\), since \(f(x)-6=(x-2)((x+1)^2+2)\). The curve is above the line exactly for \(x>2\).

Check: a cubic can meet a horizontal line once or three times. A single visible crossing on a cropped display is not proof that no others exist. Strict inequalities omit their boundary inputs.

Cubic inequalities from graphical comparisons

Cubic inequalities from graphical comparisons