Quadratic Inequalities <----- dash and solid lines?

Your first examples are AREAS, (note two variables) , and the curve is the boundary (either open or closed).

Your second pair are only curves (note one variable). If you dashed a curve then there would be no points at all!! So both curves are solid BUT in the first, the x-intercepts are open and in the second the x-intercepts are closed.

ADDENDUM, by request: A dashed anything means omit those points. Since only the points on the curve are below the x-axis (i.e. < 0), you want to INCLUDE them, so you do NOT dash them. The variable y is set = 0 (the x-axis!). So the rest are points on the curve relative to the x-axis.

If you had the equation 0 = x² + x – 20 you would have only the two points x=–5 and x=+4 on the x-axis where the curve crosses the x-axis, right? Graph y = x² + x – 20, a curve, right? Set y = 0, then two points, right? With = the points are solid. With < or with > the two points are not included. But dpn't the rest of the point *one the curve* evaluate to a number either < 0 (benartyh the x-axis) or > 0 (above the x-axis)? Dashed curves (lines) or open dots indicate a point is omitted.

So where is x² + x – 20 < 0? Answer: where the points (x, y) **on the curve** lie under the x-axis. That is, where the **curve** intersects the half-plane y < 0. A curve lying on a plane intersects that plane in a curve -- namely itself, as it is a subset of the plane.

So 0 >x² + x – 20 does not include (–5,0) and (4,0) and 0 ≥ x² + x – 20 does include those two points. That is the only difference.

Just draw the parabola. Are not all the y-coordinates points on the parabola underneath the x-axis < 0? The is one variable (in the plane) so it is one-dimensional.

As another simpler example, look: y < x is a half plane. 0<x is a half-line, right?

In the first examples, when you dash the **boundary** curve, you still have all the other points in the plane bounded by the "ghost" of the curve you left out by dashing. PLUG IN SOME VALUES. Are the included or not?

If included, then solid; excluded then open (dash or open dot, depending on the object).

ON THE OTHER HAND, did you leave out a constraint or words to indicate that this was to be a subset of the prior AREAS? Taken simply as you gave them, the second part is simply asking for points ON THE CURVE y = x² + x – 20 where y < 0.

In graphing :

y>x^2+x-20, you should use dash line to graph

y=x^2+x-20, and shade the part above the

boundary of the curve. While for

y>=x^2+x-20, you shoud use a solid line to graph

y=x^2+x-20, and shade the part above the

boundary of the curve incuding the boundary curve.

For 0>x^2+x-20, you should draw a dashed line of

y=0 from x=-5 to x=4 & a solid curve of x^2+x-20=0,

& shade the part between the line & the curve.

For 0>=x^2+x-20, you should draw a solid line of y=0

& a solid curve of x^2+x-20=0 & shade the part in bet-

ween.

y > x² + x - 20 graphs as an upward-establishing shaded-interior parabola with a dotted curve because the curve isn't blanketed contained in the equation. y ? x² + x - 20 graphs as a similar parabola except it has a reliable curve because the curve is blanketed in equation. 0 > x² + x - 20 x² + x - 20 < 0 (x + 5)(x - 4) < 0 If the manufactured from 2 elements is adverse, then in reality one component is adverse. If x + 5 < 0, x < - a million If x - 4 < 0, x < 4 (- ? < x < - 2) U (- ? < x < 4) ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 0 ? x² + x - 20 x² + x - 20 ? 0 (x + 5)(x - 4) ? 0 If x + 5 ? 0, x ? - 5 If x - 4 ? 0, x ? 4 (- ? < x ? - 5) U (- ? < x ? 4) ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯