Add and subtract algebraic fractions

These fractions name equal-sized parts. What is their sum?

Find the lowest common denominator before combining these fractions

List multiples of 4 and 6 until they first match

The given fraction is being written over 12. What must its new numerator be?

Now combine both converted numerators. Which sum is correct?

Compare the sum (x + 5)/2 + x/4 with (3x + 10)/4 and with the incorrect (2x + 5)/6. They agree at x = -4. Try x = 0 and x = 6. Values are rounded to three decimal places

Which method continues to agree with the original?

What does agreement at just one input establish?

Any common denominator can work

A lowest common denominator is efficient, but a larger common denominator is also valid if both numerators are converted correctly and the final result is reduced

1. 1Equal parts With matching denominators, combine only the numerators \(\displaystyle \frac{a}{d}+\frac{b}{d}=\frac{a+b}{d}\)
2. 2Equivalent scaling Multiply numerator and denominator by the same nonzero factor \(\displaystyle \frac{x+1}{4}=\frac{3(x+1)}{12}\)
3. 3Use the lowest when convenient For 6 and 12, using 12 avoids unnecessary arithmetic \(\displaystyle \frac{1}{6}+\frac{1}{12}=\frac{2+1}{12}\)

Use a common denominator of 12. What is this sum?

For subtraction, keep the whole second numerator in brackets. What is the combined numerator?

Check the subtraction at t = -2. Give the exact value

Substitute negative 2 into both fractions

Compare (6x + 9) - (2x + k) at x = 3. The correct result is 21 - k; changing only the sign of 2x gives the incorrect 21 + k. Start at k = 0, then try k = -4 and k = 4

Which treatment of the second bracket works for every tested constant?

Variable denominators

A variable denominator must be nonzero. For powers of the same variable, use the highest power needed to make a common denominator

1. 1Keep the original restriction Here x cannot be zero, before or after simplification \(\displaystyle x\ne0\)
2. 2Convert the smaller power Multiply both numerator and denominator by x \(\displaystyle \frac{a}{x^2}=\frac{ax}{x^3}\)
3. 3Keep the complete numerator Do not cancel a single term in a sum \(\displaystyle \frac{x+1}{x}=1+\frac{1}{x}\)

For x nonzero, choose a lowest common denominator

For nonzero x, what is the missing numerator?

For x nonzero, combine these fractions

Evaluate the given sum at x = 2. Give an exact value

Square and cube 2 before evaluating each fraction

Which value is excluded from the original expression?

For x nonzero, which form correctly combines the numerators?

Final transfer: for x nonzero, add these fractions