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Adding and subtracting

Rational expressions

Japanese school year: Math II

What you learn

Learn to add and subtract rational expressions with different denominators by finding a common denominator, and to leave the result in a form that can be simplified. It underlies partial fractions and rearranging equations. It helps to know factoring and how to find common denominators of fractions first.

Go to practice

Key points

With the same denominator, add or subtract the numerators and keep the denominator. When subtracting, put brackets around the numerator being subtracted and change its signs.

AC±BC=A±BC\frac{A}{C} \pm \frac{B}{C} = \frac{A \pm B}{C}

With different denominators, first bring them to a common denominator. Multiply each numerator by the same expression you multiplied its denominator by.

AB+CD=AD+BCBD\frac{A}{B} + \frac{C}{D} = \frac{AD + BC}{BD}

Keeping the denominators factored while bringing them together lets you see at once whether anything cancels.

1x−1−1x+1=(x+1)−(x−1)(x−1)(x+1)=2(x−1)(x+1)\frac{1}{x - 1} - \frac{1}{x + 1} = \frac{(x + 1) - (x - 1)}{(x - 1)(x + 1)} = \frac{2}{(x - 1)(x + 1)}
Worked example

Choose the expression equal to this one.

1x+1x+2\frac{1}{x} + \frac{1}{x + 2}
1x{{}\frac{1}{x}}+1x+2{{}+ \frac{1}{x + 2}}=x+2x(x+2){{}= \frac{x + 2}{x\left(x + 2\right)}}+xx(x+2){{}+ \frac{x}{x\left(x + 2\right)}}
=2x+2x(x+2){{}= \frac{2x + 2}{x\left(x + 2\right)}}
Answer: 2x+2x(x+2)\frac{2x + 2}{x\left(x + 2\right)}

Choose a set to practice.