Solve equations that contain algebraic fractions

Multiply every term by 3. What does the whole-number term 4 become?

Use x = 13 in (x + 2)/3 + 4 = 9 to get 5 + 4 = 9. In both rows, multiply the right side by m. On the left, the first row multiplies both 5 and 4; the second row forgets to multiply the 4. Change m and compare the two sides

What happens when m is greater than 1?

Clear, solve and check

Use equivalent linear-equation transformations after clearing fractions. Start with numerical denominators; variable denominators later need an extra domain check

1. 1Choose a common denominator For numerical denominators, use their least common multiple
2. 2Multiply every term Apply the same nonzero multiplier to both sides, including all whole-number terms
3. 3Keep brackets Multiply a complete numerator; subtraction applies to every term in it
4. 4Check the original Substitute your answer in the original equation, not just the cleared equation

What is the lowest common denominator of 4 and 6?

Which equation follows after multiplying every term by 12?

Solve the cleared equation exactly

When x is thirty-three fifths, what is the original left side?

Which equation correctly clears the denominators?

After expanding the left side, what constant remains?

Solve the collected equation

Extension: variable denominators

A variable denominator can be zero. Record its excluded values before clearing, and reject any candidate that violates those restrictions

1. 1Original domain Every original denominator must be nonzero \(\displaystyle x-2\ne0\Rightarrow x\ne2\)
2. 2Polynomial LCD Use a common multiple containing each distinct denominator factor to its highest required power. It need not be positive at every input
3. 3Clearing is conditional Multiplication by the LCD preserves the original equation only at allowed inputs
4. 4Check candidates A solution of the cleared equation can be forbidden in the original equation

Which value must be excluded before multiplying by x?

For x ≠ 0, which equation follows after multiplying by x?

Original equation

\(\displaystyle \frac3x+2=5\)

Cleared equation

\(\displaystyle 3+2x=5x\)

Solve and check the original equation

Which value is excluded by the original denominator?

Clear the denominator and solve

Which answer gives the solution and retains both exclusions?

Clearing gives x + 1 = 3, so the candidate is x = 2. What is the original solution set?

Thirty labels are shared equally among n + 2 groups. One group has 1 extra label added, making 6 labels. Find n.

For n = 4, how many groups were there originally?

Check the complete answer

Use the original equation and its denominator restrictions to check the complete answer. Numerical-denominator equations consolidate linear solving; the variable-denominator examples extend the same method with domain checks. In a context, report the requested quantity and its unit

1. 1Algebra Solve exactly and keep the original exclusions
2. 2Context Translate the unknown back to the requested quantity and check that it is sensible
3. 3Practice The next questions combine these steps. Enter exact fractions when needed