Solve 2/(x − 1) = 5/(x + 2)
Rational equation, worked out line by line the way a teacher would write it.
Answer
| Solution | x = 3 |
Step-by-step solution
9 steps-
1 Given\frac{2}{x - 1} = \frac{5}{x + 2}
Solve for x.
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2 Note the values that make a denominator zero — they can never be solutions
Dividing by zero is undefined, so a value that makes a denominator 0 can never be a solution, even if it turns up as an answer later.
x \neq -2,\quad x \neq 1 -
3 Cross-multiply: a/b = c/d becomes a·d = c·b
Multiplying both sides by both denominators cancels them: a/b = c/d becomes a·d = c·b. It works when each side is a single fraction.
2 \left(x + 2\right) = 5 \left(x - 1\right) -
4 Distribute: multiply each term inside the brackets
The distributive property, a(b + c) = ab + ac: the number in front of a bracket multiplies every term inside it, not just the first. A minus in front works like −1, so it flips the sign of every term inside.
4 + 2 x = -5 + 5 x -
5 Subtract 5x from both sides
An equation stays true as long as you do the same thing to both sides, like adding the same weight to both pans of a balance. Adding or subtracting the same amount on both sides moves a term to the other side, where it appears with the opposite sign.
4 - 3 x = -5Collect every term containing x on the left.
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6 Subtract 4 from both sides- 3 x = -9
Move the constant terms to the right.
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7 Divide both sides by -3
Dividing both sides by -3 undoes the multiplication by -3, so the variable is left on its own. Both sides change in the same way, so the equation stays true (dividing by 0 is the one thing that is never allowed).
x = \frac{-9}{-3} -
8 Simplifyx = 3
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9 Check\begin{aligned}x = 3:\quad 1 = 1\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| 3 | 1 | 1 | ✓ |
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