Clear the denominators first, and what is left is ordinary algebra.
Solve for x
2x3 − 56 = 38
The LCD of 3, 6 and 8 is 24. Multiplying every term by 24 gives 16x − 20 = 9, so 16x = 29 and x = 2916. B keeps the 24 as the denominator, but after clearing fractions the coefficient of x is 16.
The bottom numbers are the only thing standing in your way.
2x3 − 56 = 38
You cannot add or subtract these as they stand. Kill all three denominators in one move and the rest is ordinary algebra.
List multiples until the same number shows up in all three rows.
24 shows up in all three rows.
3 × 6 × 8 = 144 also works, but it leaves huge numbers. 24 is the smallest, so the arithmetic stays small.
Miss one term and the equation stops being true.
24 · 2x3 − 24 · 56 = 24 · 38
Whatever you do to one side you do to the other — and to every term on that side, because 24 multiplies the whole side.
A common slip is to clear the left side properly and then just erase the 8 on the right, writing 16x − 20 = 3. That gives 16x = 23 and x = 2316 — wrong, because 24 · 38 is 9, not 3. Every term on both sides gets the full × 24.
8 · 2x
16x
4 · 5
20
3 · 3
9
Put the three pieces back together.
Divide 24 by the denominator before multiplying the top. 24 ÷ 3 = 8 is easier than 48x ÷ 3.
16x − 20 = 9
no fractions left
16x = 29
+ 20 both sides
x = 2916
÷ 16 both sides
29 and 16 share no factor, so 2916 is already in lowest terms. Leave it as an improper fraction unless asked otherwise.
A fraction bar groups its whole numerator. Solve x + 43 − x − 25 = 2. The LCD is 15.
5(x + 4) − 3(x − 2) = 30
× 15: keep each numerator in brackets
5x + 20 − 3x + 6 = 30
−3 × −2 = +6
2x + 26 = 30
collect
x = 2
63 − 05 = 2 ✓
Writing 5x + 20 − 3x − 6 = 30 changes only the first sign and gives x = 8. The minus in front of the fraction belongs to the whole of 3(x − 2).
| Also asked | The move |
|---|---|
| A fraction’s denominator is 4 more than its numerator; adding 1 to both gives 23 | numerator n: n + 1n + 5 = 23, so 3(n + 1) = 2(n + 5), n = 7 — the fraction is 711 |
| Letters underneath: x + pq − x + qp = 0 | × pq: p(x + p) − q(x + q) = 0, so (p − q)x = q² − p² = −(p − q)(p + q), and x = −(p + q) |
Decimals are fractions over 10 or 100, so clear them the same way. Solve 0.3x − 1.2 = 0.15x + 0.3. The most decimal places is two, so multiply by 100.
30x − 120 = 15x + 30
× 100, every term
15x = 150 → x = 10
3 − 1.2 = 1.8 and 1.5 + 0.3 = 1.8 ✓
Word problems translate straight into a fraction equation: “a fraction of a number” is that fraction times x. Three-fourths of a number increased by 5 equals the number decreased by 2.
(34)x + 5 = x − 2
“of” is ×, “equals” is =
3x + 20 = 4x − 8 → x = 28
× 4, every term; 21 + 5 = 26 = 28 − 2 ✓
Sometimes the x terms cancel completely. What is left decides the answer.
| Equation | After × 6 | Meaning |
|---|---|---|
| x2 + x3 = 5x6 + 1 | 3x + 2x = 5x + 6 → 0 = 6 | false: no solution |
| x2 + x3 = 5x6 | 3x + 2x = 5x → 0 = 0 | always true: every x works |
When x vanishes, there is nothing left to solve for. A false statement means no number works; a true one means every number does. Both are real answer choices.
Fraction equations come as plain equations or as word problems (“three-fourths of a number increased by 5…”). Multiply every term by the LCD in one move, keep two-term numerators in brackets, and read what is asked — sometimes it is 3x − 1, not x. Checking the answer in the original takes ten seconds.
Solve for x
3x4 + 12 = x6 + 53
Multiply all four terms by the LCD 12: 9x + 6 = 2x + 20, so 7x = 14 and x = 2. Miss a single term and the answer turns into an ugly fraction — a clean result is usually a sign you caught them all.
Hint: the LCD of 4, 2, 6 and 3 is 12. This one has an x on both sides — clear the fractions first, then gather the x terms.
