Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

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ANALYTIC GEOMETRY

Example 2. Find the points satisfying simultaneously the
inequalities:
y2-2x < 0, x+.y- < 0.
The equations obtained by replacing the signs < by signs of
equality represent a parabola and a line intersecting it. The
locus of the first inequality is the interior of the parabola;
that of the second is the half-plane bounded by the line and
containing the origin. Common to these two regions is the
finite region contained between the parabola and the line;
this, then, is the locus of the two inequalities taken simultaneously.
EXERCISES
Find the locus of points whose coordinates satisfy simultaneously the following sets of inequalities. Draw a figure in
each case, and shade the region represented.

(4x-3 < 0,
1 3x+2y-6 < 0.
2x —y+3 <0,
4x_-2y+9 >0.
3x2 + 4y~- 12 > 0,
2x - 3y + 12 >0.
2x- y- 3 < 0,
7. 'x+3y-5< 0,
5x+y+3>0.

2 5x-12y +26 > 0,
2* 3x+4y-10 > 0.
f X2 + y2 < 4,.Ans.
( x-3 > 0. No locus.
2 - 7y < 0,
6. x2 y2 + >0.
I x + y2 - 1 > 0,
8. 'y>0,
2x 'y > 0.

Each of the following pairs of curves divide the plane
(minus the points on the curves) into a number of regions.
Find the pairs of simultaneous inequalities representing these
regions.

9. 5y+8=0,
10. 5x-12y+26=0,
11. 2x2+y2=8,
12. y2=2mx,

3x+8y-2 =0.
3x +4y- 10=0.
4x-3y -2= 0.
= 0.