Use the triangle inequality to establish the following inequalities: (b) ||c| – |d|| < |c – d| IPl + |이 > \p - 기 () (Note: these can be proven by cases, but they can also be deduced from the triangle inequality by choosing c or d cleverly. )

25. Describe {x : x² + 2x < 3} as an interval. Hint: Plot y = x² + 2x – 3. SOLUTION The inequality x2 + 2x < 3 is equivalent to x2 + 2x – 3 < 0. The graph of y = x2 + 2x – 3 is shown below. From this graph, it follows that x² + 2x – 3 < 0 for –3 < x < 1. Thus, the set {x : x? + 2x < 3} is equivalent to the interval (-3, 1). y -x+ 2x – 3 12 10-

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Use the Mean Value Theorem to prove the following double inequality: X 1+x 0