31. = − − 2; x2 x ( ) ( ) ( ) = + f. > g x x2 x f x g x ( ) f x − 2 ( ) = a. 0 f x − − > + − 2 2 x2 x x2 x − − 2 = 0 x2 x −2 > 0 x ( ( − 2) + 1) = < 0 x x x {x 0 The solution set is < 0} or, using x = = −1 2, x x (−∞, interval notation, 0) . {−1, Solution set: 2} . ( ) ≥ g. 1 ( ) f x b. = 0 g x − − 2 ≥ 1 x2 x + − 2 = 0 x2 x ( ( − − ≥ 3 0 x2 x + 2) − 1) = x x 0 = −2; = 1 x x {−2, (−1)2 (1) (−3) Solution set: 1} (−1) − 4 − (1) . 2 ± = x c. ( = ( f x) g x) − − = + − 2 x2 x x2 x = ± + = ± 1 1 12 1 13 2 2 2 −2 = 0 x The zeros are = 0 x ≈ 2.30 −1.30 and = 1 + 13 x {0} Solution set: . 2 2 ≈ 13 = 1 − x ( ) d. > 0 f x ( ) = − − 3 p x x2 x − − 2 > 0 x2 x ( ( , ∞ − 2) + 1) > 0 x x 2 2 −∞, , 2 2 = −1 = The zeros are and 2 . x x The solution set is ≤ or ≥ x x x ⎧⎪ 1 − 13 1 + 13 ⎫⎪ or, using ⎩⎪ ⎭⎪ 2 2 {x The solution set is < −1 or > 2} or, x x interval notation, (−∞, (2, ) ⎛ 1 − 13 ⎤ ⎡1 + 13 ⎞ interval notation, −1) ∪ ∞ −∞, ∪ ∞ . ⎜ ⎥ ⎢ ⎟ 2 2 , . ( ) e. ≤ 0 g x ⎝ ⎦⎥ ⎣⎢ ⎠ + − ≤ 2 x2 x 32. ( ) ( ) 0 = − − = − + + 6 f x x2 x g x x2 x + 1;
31. f ( x ) = x2 − x − 2; g ( x ) = x2 + x − 2 f. f ( x ) > g ( x ) a. f ( x ) = 0 x2 − x − 2 = 0 ( x − 2) ( x + 1) = 0 x = 2, x = −1 Solution set: {−1, 2} . b. g ( x ) = 0 x2 + x − 2 = 0 ( x + 2) ( x − 1) = 0 x2 − x − 2 > x2 + x − 2 −2 x > 0 x < 0 The solution set is {x x < 0} or, using interval notation, (−∞, 0) . g. f ( x ) ≥ 1 x2 − x − 2 ≥ 1 x2 − x − 3 ≥ 0 x = −2; x = 1 Solution set: {−2, 1} . − (−1) ± x = (−1)2 − 4 (1) (−3) 2 (1) c. f ( x) = g ( x) x2 − x − 2 = x2 + x − 2 −2 x = 0 = 1 ± 1 + 12 = 1 ± 13 2 2 The zeros are x = 0 Solution set: {0} . x = 1 − 13 ≈ −1.30 and x = 1 + 13 2 2 ≈ 2.30 d. f ( x ) > 0 x2 − x − 2 > 0 ( x − 2) ( x + 1) > 0 The zeros are x = −1 and x = 2 . p ( x ) = x2 − x − 3 −∞, 2 2 , 2 2 , ∞ The solution set is ⎧⎪ x x ≤ 1 − 13 or x ≥ 1 + 13 ⎫⎪ or, using ⎩⎪ 2 2 ⎭⎪ The solution set is {x x < −1 or x > 2} or, interval notation, (−∞, −1) ∪ (2, ∞ ) . interval notation, ⎛ 1 − 13 ⎤ ⎡1 + 13 ⎞ ⎜ −∞, 2 ⎥ ∪ ⎢ 2 , ∞ ⎟ . e. g ( x ) ≤ 0 ⎝ ⎦⎥ ⎣⎢ ⎠ x2 + x − 2 ≤ 0 32. f ( x ) = − x2 − x + 1; g ( x ) = − x2 + x + 6
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