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Example text

1. Transitive property: a < b and b < c STUDY TIP Once you have solved an inequality, it is a good idea to check some x-values in your solution set to see whether they satisfy the original inequality. You might also check some values outside your solution set to verify that they do not satisfy the inequality. 6 shows that when x ϭ 0 or x ϭ 2 the inequality is satisfied, but when x ϭ 4 the inequality is not satisfied. a < c aϩc < bϩd 2. Adding inequalities: a < b and c < d 3. Multiplying by a (positive) constant: a < b ac < bc, c > 0 4.

X ϩ 1͒Ϫ1͞2͑2x Ϫ 3͒5͞2 ϩ 10͑x ϩ 1͒1͞2͑2x Ϫ 3͒ 3͞2 ✓CHECKPOINT 5 Simplify the expression by factoring. ͑x ϩ 2͒1͞2͑3x Ϫ 1͒3͞2 ϩ 4͑x ϩ 2͒Ϫ1͞2͑3x Ϫ 1͒5͞2 ■ ϭ ͑x ϩ 1͒Ϫ1͞2͑2x Ϫ 3͒ 3͞2͓͑2x Ϫ 3͒ ϩ 10͑x ϩ 1͔͒ ϭ ͑x ϩ 1͒Ϫ1͞2͑2x Ϫ 3͒ 3͞2͑2x Ϫ 3 ϩ 10x ϩ 10͒ ϭ ͑x ϩ 1͒Ϫ1͞2͑2x Ϫ 3͒ 3͞2͑12x ϩ 7͒ ϭ ͑2x Ϫ 3͒ 3͞2͑12x ϩ 7͒ ͑x ϩ 1͒1͞2 16 CHAPTER 0 A Precalculus Review Example 6 shows some additional types of expressions that can occur in calculus. ͔ TECHNOLOGY > A graphing utility offers several ways to calculate rational exponents and radicals.

A B C ϩ ϩ xϩ2 xϪ3 xϩ4 A͑x Ϫ 3͒͑x ϩ 4͒ ϩ B͑x ϩ 2͒͑x ϩ 4͒ ϩ C͑x ϩ 2͒͑x Ϫ 3͒ ϭ ͑x ϩ 2͒͑x Ϫ 3͒͑x ϩ 4͒ 2 A͑x ϩ x Ϫ 12͒ ϩ B͑x 2 ϩ 6x ϩ 8͒ ϩ C͑x 2 Ϫ x Ϫ 6͒ ϭ ͑x ϩ 2͒͑x Ϫ 3͒͑x ϩ 4͒ 2 2 Ax ϩ Bx ϩ Cx 2 ϩ Ax ϩ 6Bx Ϫ Cx Ϫ 12A ϩ 8B Ϫ 6C ϭ ͑x ϩ 2͒͑x Ϫ 3͒͑x ϩ 4͒ 2 ͑A ϩ B ϩ C͒x ϩ ͑A ϩ 6B Ϫ C͒ x ϩ ͑Ϫ12A ϩ 8B Ϫ 6C͒ ϭ ͑x ϩ 2͒͑x Ϫ 3͒͑x ϩ 4͒ b. Here the least common denominator is ͑x ϩ 2͒ 2 ͑x Ϫ 1͒. A B C ϩ ϩ 2 x ϩ 2 ͑x ϩ 2͒ xϪ1 A͑x ϩ 2͒͑x Ϫ 1͒ ϩ B͑x Ϫ 1͒ ϩ C͑x ϩ 2͒ 2 ϭ ͑x ϩ 2͒ 2͑x Ϫ 1͒ A͑x 2 ϩ x Ϫ 2͒ ϩ B͑x Ϫ 1͒ ϩ C͑x 2 ϩ 4x ϩ 4͒ ϭ ͑x ϩ 2͒ 2͑x Ϫ 1͒ Ax 2 ϩ Cx 2 ϩ Ax ϩ Bx ϩ 4Cx Ϫ 2A Ϫ B ϩ 4C ϭ ͑x ϩ 2͒ 2͑x Ϫ 1͒ ͑A ϩ C͒x 2 ϩ ͑A ϩ B ϩ 4C͒x ϩ ͑Ϫ2A Ϫ B ϩ 4C͒ ϭ ͑x ϩ 2͒ 2͑x Ϫ 1͒ ✓CHECKPOINT 3 Perform each indicated addition of rational expressions.

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