The whole lesson in one picture: rebuild both fractions over a shared bottom, then add the tops. The bottoms never get added — they get matched.
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.
9.4.1 Same bottom? Combine the tops
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:
Same bottom: the denominator is the unit of counting. Combine the tops; never touch the bottom.
Watch the minus sign — the #1 trap
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.
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.
🎮 Try itCOMBINE OVER A SHARED BOTTOM
Same denominator. Flip between + and −. On −, watch the parentheses appear and the minus distribute — the naive (wrong) top is shown struck out in red.
Operation
9.4.2 Different bottoms? Build a common one
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.
Numeric warm-up → the same move with letters
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.
Worked example
Add 1x + 1x+1. The bottoms x and x+1 share no factor, so the LCD is just their product, x(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).
🎮 Try itBUILD & COMBINE (DIFFERENT BOTTOMS)
Pick a pair. Step through it: find the LCD, build each fraction up to it (missing factors highlighted), combine the tops, then reduce.
Pair
1
9.4.3 A polynomial plus a fraction: put it over 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.
Two worked examples
x1
+
1x
=
x·xx
+
1x
=
x² + 1x (x ≠ 0).
And a subtraction — mind the minus across the whole top:
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.
🎮 Try itOVER ONE: FOLD A POLYNOMIAL IN
A whole term joins a fraction. Toggle "show over 1" to see the polynomial rewritten as a fraction, then built up to the common bottom and combined.
Expression
Show over 1
9.4.4 Mixed operations & simplify-then-evaluate
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.
Worked example — a clean one
1x−1
−
1x+1.
Bottoms x−1 and x+1 share nothing, so LCD = (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.
Simplify first, then plug in the number
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.
🎮 Try itSIMPLIFY-THEN-EVALUATE RACE
Slide x. Two paths race to the same number: "plug in now" vs "simplify first, then plug in." Both agree — but the simplified path is visibly shorter. Excluded x are flagged.
x =3
★ The big ideas, in one breath
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.
What's next
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.
✎ Exercises 9.4
Combine into a single fraction in lowest terms, and state any excluded values. Try each before opening the answer.
3xx+2 + 6x+2
Show answer
Same bottom: 3x+6x+2 = 3(x+2)x+2 = 3, x ≠ −2.
5x−2x−3 − 2x+7x−3
Show answer
Distribute the minus: top = (5x−2) − (2x+7) = 5x−2−2x−7 = 3x−9. So 3x−9x−3 = 3(x−3)x−3 = 3, x ≠ 3.
LCD = (x−2)(x+2). Top = (x+2) − (x−2) = 4. So 4(x−2)(x+2) = 4x²−4, x ≠ 2, −2.
x − 2x
Show answer
x = x·xx, so = x² − 2x = x²−2x, x ≠ 0.
3 + 1x−1
Show answer
3 = 3(x−1)x−1, so = 3(x−1) + 1x−1 = 3x−2x−1, x ≠ 1.
xx²−1 + 1x+1
Show answer
Factor: x²−1 = (x−1)(x+1), so LCD = (x−1)(x+1). Build the second up by (x−1): = x + (x−1)(x−1)(x+1) = 2x−1(x−1)(x+1) = 2x−1x²−1, x ≠ 1, −1.
2xx+1 − 2x+1
Show answer
Same bottom: 2x−2x+1 = 2(x−1)x+1. The top factors but shares no factor with x+1, so this is lowest terms, x ≠ −1.
Spot the error. A student writes 4x+1x − x−5x = 4x+1 − x − 5x = 3x−4x. Right or wrong?
Show answer
Wrong! They only negated the first term of (x−5) — they wrote −x−5 instead of −x+5, forgetting to flip the −5. Distributing the minus correctly: (4x+1) − (x−5) = 4x+1 − x + 5 = 3x+6, so the answer is 3x+6x = 3(x+2)x, x ≠ 0. Their 3x−4 is the classic sign-trap error from using −x−5 instead of −x+5.
Simplify, then evaluate at x = 3: 1x−1 − 1x+1
Show answer
Simplify first: = 2(x−1)(x+1). At x = 3: 2(2)(4) = 28 = 1⁄4. (Allowed, since 3 ≠ ±1.)
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
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.