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Solving Rational Equations
Choose Course
Algebra 2
Rational Functions
Solving Rational Equations
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Solving Rational Equations 1.11 - Solution
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Return to Solving Rational Equations
We want to
solve the given rational equation
.
x
+
1
x
+
x
−
2
x
=
2
To do so, we will first determine the least
common denominator
(LCD) so that we can clear the denominators. In this case, the LCD is the product of the denominators.
x
+
1
x
+
x
−
2
x
=
2
MultEqn
LHS
⋅
(
x
+
1
)
(
x
−
2
)
=
RHS
⋅
(
x
+
1
)
(
x
−
2
)
(
x
+
1
x
+
x
−
2
x
)
(
x
+
1
)
(
x
−
2
)
=
2
(
x
+
1
)
(
x
−
2
)
Distr
Distribute
(
x
+
1
)
(
x
−
2
)
x
+
1
x
(
x
+
1
)
(
x
−
2
)
+
x
−
2
x
(
x
+
1
)
(
x
−
2
)
=
2
(
x
+
1
)
(
x
−
2
)
CancelCommonFac
Cancel out common factors
x
+
1
x
(
x
+
1
)
(
x
−
2
)
+
x
−
2
x
(
x
+
1
)
(
x
−
2
)
=
2
(
x
+
1
)
(
x
−
2
)
SimpQuot
Simplify quotient
x
(
x
−
2
)
+
x
(
x
+
1
)
=
2
(
x
+
1
)
(
x
−
2
)
Solve for
x
Distr
Distribute
x
x
2
−
2
x
+
x
2
+
x
=
2
(
x
+
1
)
(
x
−
2
)
Distr
Distribute
2
x
2
−
2
x
+
x
2
+
x
=
(
2
x
+
2
)
(
x
−
2
)
Distr
Distribute
(
k
−
2
)
x
2
−
2
x
+
x
2
+
x
=
2
x
(
x
−
2
)
+
2
(
x
−
2
)
Distr
Distribute
2
k
x
2
−
2
x
+
x
2
+
x
=
2
x
2
−
4
x
+
2
(
x
−
2
)
Distr
Distribute
2
x
2
−
2
x
+
x
2
+
x
=
2
x
2
−
4
x
+
2
x
−
4
AddSubTerms
Add and subtract terms
2
x
2
−
x
=
2
x
2
−
2
x
−
4
SubEqn
LHS
−
2
x
2
=
RHS
−
2
x
2
-
x
=
-
2
x
−
4
AddEqn
LHS
+
2
x
=
RHS
+
2
x
x
=
-
4
Finally, we will check our result to make sure that it is not an
extraneous solution
.
x
+
1
x
+
x
−
2
x
=
2
Substitute
x
=
-
4
-
4
+
1
-
4
+
-
4
−
2
-
4
=
?
2
Simplify left-hand side
AddSubTerms
Add and subtract terms
-
3
-
4
+
-
6
-
4
=
?
2
DivNegNeg
-
b
-
a
=
b
a
3
4
+
6
4
=
?
2
ReduceFrac
b
a
=
b
/
2
a
/
2
3
4
+
3
2
=
?
2
AddFrac
Add fractions
3
6
=
?
2
CalcQuot
Calculate quotient
2
=
2
✓