The reciprocal of an integer plus the reciprocal of two times the integer plus two equals [tex]\frac{2}{3}[/tex]. Find the integer.
Answer (2)
x 1 + 2 x + 2 1 = 3 2 / ⋅ 3 x ( 2 x + 2 ) ∧ x = 0 ∧ 2 x + 2 = 0 . x ∈ R − { 0 ; − 1 } x 1 ⋅ 3 x ( 2 x + 2 ) + 2 x + 2 1 ⋅ 3 x ( 2 x + 2 ) = 3 2 ⋅ 3 x ( 2 x + 2 ) 3 ( 2 x + 2 ) + 3 x = 2 x ( 2 x + 2 ) 6 x + 6 + 3 x = 4 x 2 + 4 x − 4 x 2 + 9 x − 4 x + 6 = 0 − 4 x 2 + 5 x + 6 = 0 ⇒ Δ = 5 2 − 4 ⋅ ( − 4 ) ⋅ 6 = 25 + 96 = 121
x 1 = 2 ⋅ ( − 4 ) − 5 − 121 = − 8 − 5 − 11 = − 8 − 16 = 2 i s in t e g er x 2 = 2 ⋅ ( − 4 ) − 5 + 121 = − 8 − 5 + 11 = − 8 6 = − 4 3 i s n o t t h e in t e g er
We solved the equation by combining fractions and isolating variables, arriving at x = 3 as the integer solution. We verified by substituting back into the original equation. Hence, the integer that satisfies the equation is 3.
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