1. Solving Rational Equations
Digital Lesson
Solving Rational Equations

2. are rational expressions. For example,
A rational expression is a fraction with polynomials for the numerator and denominator.
are rational expressions.
For example,
If x is replaced by a number making the denominator of a rational expression zero, the value of the rational expression is undefined.
Example: Evaluate for x = –3, 0, and 1. x
 3
undefined
undefined 9
undefined 1
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Rational Expression

3. A rational equation is an equation between rational expressions.
For example, and are rational equations.
To solve a rational equation:
1. Find the LCM of the denominators.
2. Clear denominators by multiplying both sides of the equation by the LCM.
3. Solve the resulting polynomial equation.
4. Check the solutions.
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Rational Equation

4. Examples: 1. Solve: . LCM = x – 3. 1 = x + 1 x = 0 (0) LCM = x(x – 1).
Find the LCM.
1 = x + 1
Multiply by LCM = (x – 3).
x = 0
Solve for x.
(0)
Check.
Substitute 0.
Simplify.
True.
2. Solve:
LCM = x(x – 1).
Find the LCM.
Multiply by LCM.
x – 1 = 2x
Simplify.
x = –1
Solve.
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Examples: Solve

5. In this case, the value is not a solution of the rational equation.
After clearing denominators, a solution of the polynomial equation may make a denominator of the rational equation zero.
In this case, the value is not a solution of the rational equation.
It is critical to check all solutions.
Example: Solve:
Since x2 – 1 = (x – 1)(x + 1),
LCM = (x – 1)(x + 1).
3x + 1 = x – 1
2x = – 2  x = – 1
Check.
Since – 1 makes both denominators zero, the rational equation has no solutions.
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Example: Solve

6. Find the LCD. Solve: 2 + 1 = 4 x 3 x 3x 3x ∙ 2 + 3x ∙ 1 = 3x ∙ 4 x 3 x
Multiply each term by the LCD.
6 + x = 12
Simplify
x = 6

7. Find the LCD. Solve: 6 + x = 4 x 2 2x 2x ∙ 6 + 2x ∙ x = 2x ∙ 4 x 2
Multiply each term by the LCD.
12 + x2 = 8x
Simplify
x2 – 8x + 12 = 0
(x – 6)(x – 2) = 0
x = 6, 2

8. Solve: 5 = y y + 2 3 Cross Multiply 15 = y(y + 2) 15 = y2 + 2y
Simplify
Write in standard form.
Factor
Solve

9. Solve: 3 = 2 . x + 5 x + 1 Cross Multiply 3(x + 1) = 2(x + 5)
Simplify
Combine Like-Terms to solve.

10. Example: Solve: . x2 – 8x + 15 = (x – 3)(x – 5) x(x – 5) = – 6
Factor.
The LCM is (x – 3)(x – 5).
Original Equation.
x(x – 5) = – 6
Polynomial Equation.
x2 – 5x + 6 = 0
Simplify.
(x – 2)(x – 3) = 0
Factor.
Check. x = 2 is a solution.
x = 2 or x = 3
Check. x = 3 is not a solution since both sides would be undefined.
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Example: Solve