Algebra Examples

Solve the Rational Equation for x 1/(3x-2)+1=3/(3x+2)
Step 1
Subtract from both sides of the equation.
Step 2
Find the LCD of the terms in the equation.
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Step 2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 2.2
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
Step 2.3
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 2.4
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 2.5
The factor for is itself.
occurs time.
Step 2.6
The factor for is itself.
occurs time.
Step 2.7
The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.
Step 3
Multiply each term in by to eliminate the fractions.
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Step 3.1
Multiply each term in by .
Step 3.2
Simplify the left side.
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Step 3.2.1
Cancel the common factor of .
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Step 3.2.1.1
Cancel the common factor.
Step 3.2.1.2
Rewrite the expression.
Step 3.3
Simplify the right side.
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Step 3.3.1
Simplify each term.
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Step 3.3.1.1
Cancel the common factor of .
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Step 3.3.1.1.1
Factor out of .
Step 3.3.1.1.2
Cancel the common factor.
Step 3.3.1.1.3
Rewrite the expression.
Step 3.3.1.2
Apply the distributive property.
Step 3.3.1.3
Multiply by .
Step 3.3.1.4
Multiply by .
Step 3.3.1.5
Expand using the FOIL Method.
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Step 3.3.1.5.1
Apply the distributive property.
Step 3.3.1.5.2
Apply the distributive property.
Step 3.3.1.5.3
Apply the distributive property.
Step 3.3.1.6
Combine the opposite terms in .
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Step 3.3.1.6.1
Reorder the factors in the terms and .
Step 3.3.1.6.2
Subtract from .
Step 3.3.1.6.3
Add and .
Step 3.3.1.7
Simplify each term.
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Step 3.3.1.7.1
Rewrite using the commutative property of multiplication.
Step 3.3.1.7.2
Multiply by by adding the exponents.
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Step 3.3.1.7.2.1
Move .
Step 3.3.1.7.2.2
Multiply by .
Step 3.3.1.7.3
Multiply by .
Step 3.3.1.7.4
Multiply by .
Step 3.3.1.8
Apply the distributive property.
Step 3.3.1.9
Multiply by .
Step 3.3.1.10
Multiply by .
Step 3.3.2
Add and .
Step 4
Solve the equation.
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Step 4.1
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 4.2
Move all terms containing to the left side of the equation.
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Step 4.2.1
Subtract from both sides of the equation.
Step 4.2.2
Subtract from .
Step 4.3
Move all terms to the left side of the equation and simplify.
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Step 4.3.1
Subtract from both sides of the equation.
Step 4.3.2
Subtract from .
Step 4.4
Use the quadratic formula to find the solutions.
Step 4.5
Substitute the values , , and into the quadratic formula and solve for .
Step 4.6
Simplify.
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Step 4.6.1
Simplify the numerator.
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Step 4.6.1.1
Raise to the power of .
Step 4.6.1.2
Multiply .
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Step 4.6.1.2.1
Multiply by .
Step 4.6.1.2.2
Multiply by .
Step 4.6.1.3
Subtract from .
Step 4.6.1.4
Rewrite as .
Step 4.6.1.5
Rewrite as .
Step 4.6.1.6
Rewrite as .
Step 4.6.1.7
Rewrite as .
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Step 4.6.1.7.1
Factor out of .
Step 4.6.1.7.2
Rewrite as .
Step 4.6.1.8
Pull terms out from under the radical.
Step 4.6.1.9
Move to the left of .
Step 4.6.2
Multiply by .
Step 4.6.3
Simplify .
Step 4.7
The final answer is the combination of both solutions.