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# Rational equations

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How to Solve Rational Equations

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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