Stitz-Zeager_College_Algebra_e-book

In english it says that if we want to prove that a

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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...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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