Match the bottoms first — then combine the tops, and mind the minus sign.
Point 3 of 4 in this lesson: 9.4.3 A polynomial plus a fraction: put it over 1
You already know how to add 1⁄2 + 1⁄3. You do not write 2⁄5 — you find a common bottom, 6, rebuild each fraction over it, and get 3⁄6 + 2⁄6 = 5⁄6. Adding rational expressions is the very same dance. The only twist: the bottoms now hide a letter, so "find a common bottom" means factor and build an LCD, and a stray minus sign can quietly wreck the whole thing.
By the end of this lesson you'll be able to: add and subtract fractions that already share a bottom (combine the tops, keep the bottom); add and subtract fractions with different bottoms (find the LCD, build up, then combine); fold a plain polynomial like x into a fraction by writing it over 1; and run a mixed expression in the right order. Throughout, we keep the same color code as the rest of Stage 9: the numerator (top) is amber, the denominator (bottom) is blue, a cancelled common factor is green, and an excluded value — an x that would make a bottom 0 — is red.
When two fractions already share a denominator, addition is almost no work. Adding three eighths and two eighths gives five eighths — the eighths don't change, only how many you have. In symbols:
For subtraction, the minus belongs to the whole second top, not just its first term. Put parentheses around it, then distribute:
2x+1x−1 − x+3x−1 = (2x+1) − (x+3)x−1 = x−2x−1
Because (2x+1) − (x+3) = 2x+1 − x − 3 = x − 2. The wrong move forgets the parentheses: 2x+1 − x + 3 = x + 4 — that flips the sign of the +3 and lands on the wrong answer.
3xx+2 + 6x+2 = 3x + 6x+2 = 3(x+2)x+2 = 3 (with x ≠ −2).
Always check whether the combined top shares a factor with the bottom — here (x+2) cancels and the whole thing collapses to a plain 3. The restriction x ≠ −2 stays, because it came from the original bottom.
If the bottoms don't match, you can't combine yet — exactly like 1⁄2 + 1⁄3. The fix is the LCD from 9.2: factor each bottom, take every distinct factor to its highest power, then use the fundamental property to build each fraction up to that shared bottom. Only then do you combine the tops.
1⁄2 + 1⁄3: LCD is 6, so 1⁄2 = 3⁄6 and 1⁄3 = 2⁄6, giving 3⁄6 + 2⁄6 = 5⁄6. Notice you multiplied each fraction by a clever form of 1 (3⁄3, then 2⁄2) to reach the common bottom. With letters, that clever 1 is built from the missing factors.
Add 1x + 1x+1. The bottoms x and x+1 share no factor, so the LCD is just their product, x(x+1).
1·(x+1)x(x+1) + 1·xx(x+1) = (x+1) + xx(x+1) = 2x+1x(x+1)
The top 2x+1 shares no factor with x(x+1), so this is already in lowest terms. Restrictions: x ≠ 0 and x ≠ −1.
When the bottoms do share factors, factor first so you don't over-build. For example, with bottoms x²−1 = (x−1)(x+1) and x+1, the LCD is just (x−1)(x+1) — you don't need a second copy of (x+1).
What about x + 1⁄x? The x is not a fraction — until you remember that every quantity is itself over 1. Write x as x1, and now you have two fractions to combine the usual way.
x1 + 1x = x·xx + 1x = x² + 1x (x ≠ 0).
And a subtraction — mind the minus across the whole top:
2 − 3x+1 = 2(x+1)x+1 − 3x+1 = 2(x+1) − 3x+1 = 2x − 1x+1 (x ≠ −1).
A common slip is to write x + 1⁄x = (x+1)⁄x. That secretly multiplied x by 1 instead of by x⁄x. To put x over the bottom x you must scale it by x⁄x, giving x·x = x² on top — not x.
When several operations meet, the order is the same one you've used since arithmetic: parentheses → powers → × and ÷ (left to right) → + and −. Do any multiplying or dividing (flipping for each ÷, cancelling first), and only then add or subtract.
1x−1 − 1x+1. Bottoms x−1 and x+1 share nothing, so LCD = (x−1)(x+1):
(x+1) − (x−1)(x−1)(x+1) = x+1 − x+1(x−1)(x+1) = 2(x−1)(x+1)
The two x's cancel and the top collapses to 2 — a satisfying, fully reduced result. Restrictions: x ≠ 1, x ≠ −1.
When a problem asks you to evaluate an expression at some x, simplify to lowest terms before you substitute. It's far less arithmetic, and it sidesteps blowups. Take the result above at x = 3.
| Plug in early (messy) | Simplify first (easy) |
|---|---|
| 1⁄(3−1) − 1⁄(3+1) = 1⁄2 − 1⁄4 = 2⁄4 − 1⁄4 = 1⁄4 |
2 ⁄ ((3−1)(3+1)) = 2 ⁄ (2·4) = 2⁄8 = 1⁄4 |
Both reach 1⁄4 — but the simplified form is one short division, and on harder expressions the gap is enormous.
To add or subtract rational expressions: match the bottoms, then combine the tops. If the bottoms already agree, just combine the tops over the shared bottom and reduce. If they differ, build the LCD (factor each bottom, take every distinct factor to its highest power), rebuild each fraction up to it, and only then combine. You never add bottoms — you match them. For subtraction, wrap the second top in parentheses and distribute the minus to every term. A bare polynomial joins the party by going over 1. And to evaluate, simplify first, then substitute.
Everything so far has been about simplifying expressions. In 9.5 we add an equals sign and solve — clearing all the denominators at once, then checking for the fake "extraneous" roots that clearing can invent. The minus-sign care and the excluded values you've practiced here are exactly what keeps that check honest.
Combine into a single fraction in lowest terms, and state any excluded values. Try each before opening the answer.
Six questions to lock it in. Tap the answer you think is right.
This lesson develops CCSS A-APR.D.6 and A-APR.D.7 (rewrite and add/subtract rational expressions), leaning on A-SSE.A.2 (using structure — factoring bottoms to build the LCD) and echoing the numeric foundation of 7.EE.A.1 (adding linear expressions with common denominators). The unifying skill is recognizing that a rational expression behaves exactly like a numeric fraction, with the added vigilance that its bottom can be zero.
The #1 misconception is the subtraction sign error: distributing the minus to only the first term of the second numerator, writing … − x + 3 instead of … − x − 3 — and, close behind, adding the denominators (1⁄2 + 1⁄3 ⇒ 2⁄5). The antidote: say it out loud every time — "subtract the whole top: put it in parentheses first, then distribute," and "you never add bottoms, you match them." Insisting on visible parentheses around every subtracted numerator, and a written LCD before any combining, eliminates the great majority of errors.
Continue: 9.1 Meeting the Rational Expression · 9.2 Reducing & Common Denominators · 9.3 Multiplying, Dividing & Powers · 9.5 Rational Equations