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a. \({4 \over {x + 2}} + {2 \over {x – 2}} + {{5x – 6} \over {4 – {x^2}}}\)
b. \({{1 – 3x} \over {2x}} + {{3x – 2} \over {2x – 1}} + {{3x – 2} \over {2x – 4{x^2}}}\)
c. \({1 \over {{x^2} + 6x + 9}} + {1 \over {6x – {x^2} – 9}} + {x \over {{x^2} – 9}}\)
d. \({{{x^2} + 2} \over {{x^3} – 1}} + {2 \over {{x^2} + x + 1}} + {1 \over {1 – x}}\)
e. \({x \over {x – 2y}} + {x \over {x + 2y}} + {{4xy} \over {4{y^2} – {x^2}}}\)
a. \({4 \over {x + 2}} + {2 \over {x – 2}} + {{5x – 6} \over {4 – {x^2}}}\) \( = {4 \over {x + 2}} + {2 \over {x – 2}} + {{6 – 5x} \over {\left( {x + 2} \right)\left( {x – 2} \right)}}\)
\(\eqalign{ & = {{4\left( {x – 2} \right)} \over {\left( {x + 2} \right)\left( {x – 2} \right)}} + {{2\left( {x + 2} \right)} \over {\left( {x + 2} \right)\left( {x – 2} \right)}} + {{6 – 5x} \over {\left( {x + 2} \right)\left( {x – 2} \right)}} = {{4x – 8 + 2x + 4 + 6 – 5x} \over {\left( {x + 2} \right)\left( {x – 2} \right)}} \cr & = {{x + 2} \over {\left( {x + 2} \right)\left( {x – 2} \right)}} = {1 \over {x – 2}} \cr} \)
b. \({{1 – 3x} \over {2x}} + {{3x – 2} \over {2x – 1}} + {{3x – 2} \over {2x – 4{x^2}}}\) \( = {{1 – 3x} \over {2x}} + {{3x – 2} \over {2x – 1}} + {{2 – 3x} \over {2x\left( {2x – 1} \right)}}\)
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\(\eqalign{ & = {{\left( {1 – 3x} \right)\left( {2x – 1} \right)} \over {2x\left( {2x – 1} \right)}} + {{\left( {3x – 2} \right).2x} \over {2x\left( {2x – 1} \right)}} + {{2 – 3x} \over {2x\left( {2x – 1} \right)}} \cr & = {{2x – 1 – 6{x^2} + 3x + 6{x^2} – 4x + 2 – 3x} \over {2x\left( {2x – 1} \right)}} = {{1 – 2x} \over {2x\left( {2x – 1} \right)}} = {{ – \left( {2x – 1} \right)} \over {2x\left( {2x – 1} \right)}} = {{ – 1} \over {2x}} \cr} \)
c. \({1 \over {{x^2} + 6x + 9}} + {1 \over {6x – {x^2} – 9}} + {x \over {{x^2} – 9}}\)\( = {1 \over {{{\left( {x + 3} \right)}^2}}} + {{ – 1} \over {{{\left( {x – 3} \right)}^2}}} + {x \over {\left( {x + 3} \right)\left( {x – 3} \right)}}\)
\(\eqalign{ & = {{{{\left( {x – 3} \right)}^2}} \over {{{\left( {x + 3} \right)}^2}{{\left( {x – 3} \right)}^2}}} + {{ – {{\left( {x + 3} \right)}^2}} \over {{{\left( {x + 3} \right)}^2}{{\left( {x – 3} \right)}^2}}} + {{x\left( {x + 3} \right)\left( {x – 3} \right)} \over {{{\left( {x + 3} \right)}^2}{{\left( {x – 3} \right)}^2}}} \cr & = {{{x^2} – 6x + 9 – {x^2} – 6x – 9 + {x^3} – 9x} \over {{{\left( {x + 3} \right)}^2}{{\left( {x – 3} \right)}^2}}} = {{{x^3} – 21x} \over {{{\left( {x + 3} \right)}^2}{{\left( {x – 3} \right)}^2}}} \cr} \)
d. \({{{x^2} + 2} \over {{x^3} – 1}} + {2 \over {{x^2} + x + 1}} + {1 \over {1 – x}}\)\( = {{{x^2} + 2} \over {\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} + {2 \over {{x^2} + x + 1}} + {{ – 1} \over {x – 1}}\)
\(\eqalign{ & = {{{x^2} + 2} \over {\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} + {{2\left( {x – 1} \right)} \over {\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} + {{ – \left( {{x^2} + x + 1} \right)} \over {\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} \cr & = {{{x^2} + 2 + 2x – 2 – {x^2} – x – 1} \over {\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} = {{x – 1} \over {\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} = {1 \over {{x^2} + x + 1}} \cr} \)
e. \({x \over {x – 2y}} + {x \over {x + 2y}} + {{4xy} \over {4{y^2} – {x^2}}}\)\( = {x \over {x – 2y}} + {x \over {x + 2y}} + {{ – 4xy} \over {\left( {x + 2y} \right)\left( {x – 2y} \right)}}\)
\(\eqalign{ & = {{x\left( {x + 2y} \right)} \over {\left( {x – 2y} \right)\left( {x + 2y} \right)}} + {{x\left( {x – 2y} \right)} \over {\left( {x – 2y} \right)\left( {x + 2y} \right)}} + {{ – 4xy} \over {\left( {x – 2y} \right)\left( {x + 2y} \right)}} \cr & = {{{x^2} + 2xy + {x^2} – 2xy – 4xy} \over {\left( {x – 2y} \right)\left( {x + 2y} \right)}} = {{2{x^2} – 4xy} \over {\left( {x – 2y} \right)\left( {x + 2y} \right)}} = {{2x\left( {x – 2y} \right)} \over {\left( {x – 2y} \right)\left( {x + 2y} \right)}} \cr & = {{2x} \over {x + 2y}} \cr} \)