**Unformatted text preview: **lt; 4 − y 2
We put this technique to good use in the following example.
Example 8.7.5. Sketch the solution to the following nonlinear inequalities in the plane.
1. y 2 − 4 ≤ x < y + 2 2. x2 x2 + y 2 ≥ 4
− 2x + y 2 − 2y ≤ 0 Solution.
1. The inequality y 2 − 4 ≤ x < y + 2 is a compound inequality. It translates as y 2 − 4 ≤ x
and x < y + 2. As usual, we solve each inequality and take the set theoretic intersection
to determine the region which satisﬁes both inequalities. To solve y 2 − 4 ≤ x, we write
y 2 − x − 4 ≤ 0. The curve y 2 − x − 4 = 0 describes a parabola since exactly one of the
variables is squared. Rewriting this in standard form, we get y 2 = x + 4 and we see that the
vertex is (−4, 0) and the parabola opens to the right. Using the test points (−5, 0) and (0, 0),
we ﬁnd that the solution to the inequality includes the region to the right of, or ‘inside’, the
7
The theory behind why all this works is, surprisingly, the same theory which guarantees that sign diagrams work
the way they do - continuity and...

View
Full
Document