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Solving
Rational Equations
By: Ms. Ruth Good
Algebra 2
Rational Equations
• Def: Equations with a variable in
the denominator
• Example:
1
2
1
=
+
+
x
x
x
Steps for Solving:
Find LCD of all terms in equation.
Multiply both sides of equation by
LCD.
(This results in a new equ...
Definition
• Some answers will
not work in the
original equation
because they will
make the
denominator equal
zero. These ...
Example 1: Solve
Find LCD. LCD = 2x
xx
12
2
13
=−
x
xx
x 2
12
2
13
2 





=





− Multiply both sides by 2x....
Checking Ex. 1:
Substitute in x = -18.
18
12
2
1
18
3
−
=−
−
Get common denominators.
18
12
18
12
18
12
18
9
18
3
−
=
−
−
...
Example 2: Solve
LCD = x + 1
1
5
4
1
5
+
−=
+ xx
x
( ) ( )1
1
5
4
1
5
1 +





+
−=





+
+ x
xx
x
x
5445 −+=...
Example 3: Solve
Factor denominator of second fraction.1
4
6
2
23
2
+
−
=
−
−
xx
x
1
)2)(2(
6
2
23
+
−+
=
−
−
xxx
x
LCD = ...
Example 4: Solve
4
1
4
3
2
+
=
+ xxx
What is different about this equation?
3x + 12 = x2
+ 4x Cross-Multiply
0 = x2
+ x – ...
Example 4: Solve
4
1
4
3
2
+
=
+ xxx
What is different about this equation?
3x + 12 = x2
+ 4x Cross-Multiply
0 = x2
+ x – ...
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Rational equations

  1. 1. Solving Rational Equations By: Ms. Ruth Good Algebra 2
  2. 2. Rational Equations • Def: Equations with a variable in the denominator • Example: 1 2 1 = + + x x x
  3. 3. Steps for Solving: Find LCD of all terms in equation. Multiply both sides of equation by LCD. (This results in a new equation which may not equal the original equation.) Solve resulting equation. Check answers in original equation.
  4. 4. Definition • Some answers will not work in the original equation because they will make the denominator equal zero. These are called extraneous roots.
  5. 5. Example 1: Solve Find LCD. LCD = 2x xx 12 2 13 =− x xx x 2 12 2 13 2       =      − Multiply both sides by 2x. 246 =− x Solve. 18 18 −= =− x x Now we must check…
  6. 6. Checking Ex. 1: Substitute in x = -18. 18 12 2 1 18 3 − =− − Get common denominators. 18 12 18 12 18 12 18 9 18 3 − = − − =− − It checks!!! X = -18
  7. 7. Example 2: Solve LCD = x + 1 1 5 4 1 5 + −= + xx x ( ) ( )1 1 5 4 1 5 1 +      + −=      + + x xx x x 5445 −+= xx 1−=x Solve. 0 5 4 0 5 −= − Doesn’t check. No Solution!! Check:
  8. 8. Example 3: Solve Factor denominator of second fraction.1 4 6 2 23 2 + − = − − xx x 1 )2)(2( 6 2 23 + −+ = − − xxx x LCD = (x+2)(x-2) )2)(2(1 )2)(2( 6 2 23 )2)(2( −+      + −+ =      − − −+ xx xxx x xx ( )( ) )2)(2(6223 +−+=+− xxxx 1,3 0)1)(3(2 0)32(2 0642 464263 2 2 22 −= =−+ =−+ =−+ −+=−−+ x xx xx xx xxxx Solve. We still need to check. X = -3, 1
  9. 9. Example 4: Solve 4 1 4 3 2 + = + xxx What is different about this equation? 3x + 12 = x2 + 4x Cross-Multiply 0 = x2 + x – 12 Solve 0 = (x + 4)(x – 3) x = -4, 3 Check X = 3
  10. 10. Example 4: Solve 4 1 4 3 2 + = + xxx What is different about this equation? 3x + 12 = x2 + 4x Cross-Multiply 0 = x2 + x – 12 Solve 0 = (x + 4)(x – 3) x = -4, 3 Check X = 3

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