H Algebra 2/Trig · Unit 4
Solve rational equations by clearing denominators, then reject any divide-by-zero value.
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Clear the denominators
Factor every denominator and write the LCD.
Multiply both sides by the LCD. The equation becomes a polynomial equation.
Divide-by-zero
A value that makes a denominator 0 cannot be a solution.
Write those values before you solve.
Check every candidate
A root of the polynomial can still be a divide-by-zero value.
Reject that value. It is not a solution.
No solution
If every candidate is excluded, the equation has no solution.
A nonzero number set equal to 0 also means no solution.
One fraction, or a fraction equal to a number
The LCD is x.
Divide-by-zero values cannot be solutions.
x ≠ 0
Multiply both sides by the LCD.
Left side:
4x² + 3x
Right side:
10
Move every term to one side.
4x² + 3x − 10 = 0
Factor.
(4x − 5)(x + 2) = 0
4x − 5 = 0
x = 54
x + 2 = 0
x = −2
x = 54 or x = −2
Answer: x = 54 or x = −2
The LCD is (x − 2).
Divide-by-zero values cannot be solutions.
x ≠ 2
Multiply both sides by the LCD.
Left side:
x + 6
Right side:
4x − 8
Move every term to one side.
14 − 3x = 0
−3x = −14
x = 143
Answer: x = 143
The LCD is x.
Divide-by-zero values cannot be solutions.
x ≠ 0
Multiply both sides by the LCD.
Left side:
2x² − 7x
Right side:
−3
Move every term to one side.
2x² − 7x + 3 = 0
Factor.
(x − 3)(2x − 1) = 0
x − 3 = 0
x = 3
2x − 1 = 0
x = 12
x = 3 or x = 12
Answer: x = 3 or x = 12
The LCD is (x + 2).
Divide-by-zero values cannot be solutions.
x ≠ −2
Multiply both sides by the LCD.
Left side:
3x − 1
Right side:
2x + 4
Move every term to one side.
x − 5 = 0
x = 5
Answer: x = 5
The LCD is x.
Divide-by-zero values cannot be solutions.
x ≠ 0
Multiply both sides by the LCD.
Left side:
x² + 6x
Right side:
7
Move every term to one side.
x² + 6x − 7 = 0
Factor.
(x − 1)(x + 7) = 0
x − 1 = 0
x = 1
x + 7 = 0
x = −7
x = 1 or x = −7
Answer: x = 1 or x = −7
The LCD is (x + 1).
Divide-by-zero values cannot be solutions.
x ≠ −1
Multiply both sides by the LCD.
Left side:
x − 4
Right side:
3x + 3
Move every term to one side.
−2x − 7 = 0
−2x = 7
x = −72
Answer: x = −72
A polynomial term and a rational term
The LCD is (x − 3).
Divide-by-zero values cannot be solutions.
x ≠ 3
Multiply both sides by the LCD.
Left side:
x
x² − 3x
Combine.
x² − 2x
Right side:
6 − x
Move every term to one side.
x² − x − 6 = 0
Factor.
(x − 3)(x + 2) = 0
x − 3 = 0
x = 3
x + 2 = 0
x = −2
x = 3 is a divide-by-zero value, so it is not a solution.
x = −2
Answer: x = −2
The LCD is (x + 2).
Divide-by-zero values cannot be solutions.
x ≠ −2
Multiply both sides by the LCD.
Left side:
x
2x + 4
Combine.
3x + 4
Right side:
x + 8
Move every term to one side.
2x − 4 = 0
2x = 4
x = 42
x = 2
Answer: x = 2
The LCD is (x − 1).
Divide-by-zero values cannot be solutions.
x ≠ 1
Multiply both sides by the LCD.
Left side:
x
3x − 3
Combine.
4x − 3
Right side:
x + 6
Move every term to one side.
3x − 9 = 0
3x = 9
x = 93
x = 3
Answer: x = 3
The LCD is (x − 2).
Divide-by-zero values cannot be solutions.
x ≠ 2
Multiply both sides by the LCD.
Left side:
3
2x − 4
Combine.
2x − 1
Right side:
x + 4
Move every term to one side.
x − 5 = 0
x = 5
Answer: x = 5
The LCD is (x + 1).
Divide-by-zero values cannot be solutions.
x ≠ −1
Multiply both sides by the LCD.
Left side:
x
2x + 2
Combine.
3x + 2
Right side:
3x + 4
Move every term to one side.
−2 = 0
A nonzero number cannot equal 0.
No solution.
Answer: no solution
Factor x² − 4.
(x − 2)(x + 2)
The LCD is (x − 2)(x + 2).
Divide-by-zero values cannot be solutions.
x ≠ −2 and x ≠ 2
Multiply both sides by the LCD.
Left side:
x + 2
x − 2
Combine.
2x
Right side:
4
Move every term to one side.
2x − 4 = 0
2x = 4
x = 42
x = 2
x = 2 is a divide-by-zero value, so it is not a solution.
No solution.
Answer: no solution
An opposite factor, or no solution
Factor 2 − x.
−(x − 2)
The LCD is (x − 2).
Divide-by-zero values cannot be solutions.
x ≠ 2
Multiply both sides by the LCD.
Left side:
−x − 1
Right side:
3x − 6
Move every term to one side.
5 − 4x = 0
−4x = −5
x = 54
Answer: x = 54
Factor x² − 1.
(x − 1)(x + 1)
The LCD is (x − 1)(x + 1).
Divide-by-zero values cannot be solutions.
x ≠ −1 and x ≠ 1
Multiply both sides by the LCD.
Left side:
x + 1
x − 1
Combine.
2x
Right side:
2
Move every term to one side.
2x − 2 = 0
2x = 2
x = 22
x = 1
x = 1 is a divide-by-zero value, so it is not a solution.
No solution.
Answer: no solution
Factor 4 − x.
−(x − 4)
The LCD is (x − 4).
Divide-by-zero values cannot be solutions.
x ≠ 4
Multiply both sides by the LCD.
Left side:
−x − 2
Right side:
2x − 8
Move every term to one side.
6 − 3x = 0
−3x = −6
x = 63
x = 2
Answer: x = 2
Factor 3 − x.
−(x − 3)
Factor 3 − x.
−(x − 3)
The LCD is (x − 3).
Divide-by-zero values cannot be solutions.
x ≠ 3
Multiply both sides by the LCD.
Left side:
−x
x − 3
Combine.
−3
Right side:
−4
Move every term to one side.
1 = 0
A nonzero number cannot equal 0.
No solution.
Answer: no solution
Factor 5 − x.
−(x − 5)
The LCD is (x − 5).
Divide-by-zero values cannot be solutions.
x ≠ 5
Multiply both sides by the LCD.
Left side:
1 − x
Right side:
2x − 10
Move every term to one side.
11 − 3x = 0
−3x = −11
x = 113
Answer: x = 113
The LCD is (x − 1)(x + 1).
Divide-by-zero values cannot be solutions.
x ≠ −1 and x ≠ 1
Multiply both sides by the LCD.
Left side:
x² + 5x + 4
x² − x
Combine.
2x² + 4x + 4
Right side:
2x² − 2
Move every term to one side.
4x + 6 = 0
4x = −6
x = −64
x = −32
Answer: x = −32
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