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Add and Subtract Rational Expressions

Add and subtract rational expressions with a common denominator

Add and Subtract Rational Expressions with a Common Denominator

What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.

It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.

To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result over the common denominator.

We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.

Remember, too, we do not allow values that would make the denominator zero. What value of x should be excluded in the next example?

Example

Try it.

Add: \(\frac{11x+28}{x+4}+\frac{{x}^{2}}{x+4}.\)

Solution

Since the denominator is \(x+4,\) we must exclude the value \(x=-4.\)

	\(\ \frac{11x+28}{x+4}+\frac{{x}^{2}}{x+4},\ x\ne \text{-}4\)
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\ \frac{11x+28+{x}^{2}}{x+4}\)
Write the degrees in descending order.	\(\ \frac{{x}^{2}+11x+28}{x+4}\)
Factor the numerator.	\(\ \frac{(x+4)(x+7)}{x+4}\)
Simplify by removing common factors.	\(\ \frac{(x+4)(x+7)}{x+4}\)
Simplify.	\(\ x+7\)

The expression simplifies to \(x+7\) but the original expression had a denominator of \(x+4\) so \(x\ne \text{-}4.\)

To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator. Be careful of the signs when you subtract a binomial or trinomial.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Add and Subtract Rational Expressions Whose Denominators are Opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by \(\frac{-1}{-1}.\)

Let’s see how this works.


Multiply the second fraction by \(\frac{-1}{-1}.\)
The denominators are the same.
Simplify.

Be careful with the signs as you work with the opposites when the fractions are being subtracted.

Example

Try it.

Subtract: \(\frac{{m}^{2}-6m}{{m}^{2}-1}-\frac{3m+2}{1-{m}^{2}}.\)

Solution


The denominators are opposites, so multiply the second fraction by \(\frac{-1}{-1}.\)
Simplify the second fraction.
The denominators are the same. Subtract the numerators.
Distribute.
Combine like terms.
Factor the numerator and denominator.
Simplify by removing common factors.
Simplify.

Find the Least Common Denominator of Rational Expressions

When we add or subtract rational expressions with unlike denominators, we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.

Let’s look at this example: \(\frac{7}{12}+\frac{5}{18}.\) Since the denominators are not the same, the first step was to find the least common denominator (LCD).

To find the LCD of the fractions, we factored 12 and 18 into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD.

When we add numerical fractions, once we found the LCD, we rewrote each fraction as an equivalent fraction with the LCD by multiplying the numerator and denominator by the same number. We are now ready to add.

We do the same thing for rational expressions. However, we leave the LCD in factored form.

Remember, we always exclude values that would make the denominator zero. What values of \(x\) should we exclude in this next example?

Example

Try it.

ⓐ Find the LCD for the expressions \(\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}\) and ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

Solution

ⓐ

Find the LCD for \(\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}.\)
Factor each denominator completely, lining up common factors. Bring down the columns.
Write the LCD as the product of the factors.

ⓑ


Factor each denominator.
Multiply each denominator by the ‘missing’ LCD factor and multiply each numerator by the same factor.
Simplify the numerators.

Add and Subtract Rational Expressions with Unlike Denominators

Now we have all the steps we need to add or subtract rational expressions with unlike denominators.

How to Add Rational Expressions with Unlike Denominators

Try it.

Add: \(\frac{3}{x-3}+\frac{2}{x-2}.\)

Solution

The steps used to add rational expressions are summarized here.

Avoid the temptation to simplify too soon. In the example above, we must leave the first rational expression as \(\frac{3x-6}{(x-3)(x-2)}\) to be able to add it to \(\frac{2x-6}{(x-2)(x-3)}.\) Simplify only after you have combined the numerators.

Example

Try it.

Add: \(\frac{8}{{x}^{2}-2x-3}+\frac{3x}{{x}^{2}+4x+3}.\)

Solution


Do the expressions have a common denominator?	No.
Rewrite each expression with the LCD.
\(\begin{array}{llll} \\ \\ \text{Find the LCD.} & & & \begin{array}{l}\ {x}^{2}-2x-3=(x+1)(x-3) \\ \underset{______________________________}{{x}^{2}+4x+3=(x+1)\ (x+3)}\ \\ \\ \text{LCD}\ =(x+1)(x-3)(x+3)\end{array}\end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Add the rational expressions.
Simplify the numerator.
	The numerator is prime, so there are no common factors.

The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.

Example

Try it.

Subtract: \(\frac{8y}{{y}^{2}-16}-\frac{4}{y-4}.\)

Solution


Do the expressions have a common denominator?	No.
Rewrite each expression with the LCD.
\(\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}{y}^{2}-16=(y-4)(y+4) \\ \underset{____________}{\text{}y-4\ =y-4} \\ \text{LCD}\ =(y-4)(y+4)\end{array}\end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Subtract the rational expressions.
Simplify the numerator.
Factor the numerator to look for common factors.
Remove common factors
Simplify.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Add and subtract rational functions

To add or subtract rational functions, we use the same techniques we used to add or subtract polynomial functions.

Example

Try it.

Find \(R(x)=f(x)-g(x)\) where \(f(x)=\frac{x+5}{x-2}\) and \(g(x)=\frac{5x+18}{{x}^{2}-4}.\)

Solution


Substitute in the functions \(f(x),\)\(g(x).\)
Factor the denominators.
Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
\(\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}x-2\ =(x-2) \\ \underset{___________________}{{x}^{2}-4=(x-2)(x+2)} \\ \text{LCD}=(x-2)(x+2)\end{array}\end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Write as one rational expression.
Simplify.

Factor the numerator, and remove common factors.
Simplify.

Key Concepts

Rational Expression Addition and Subtraction
If p, q, and r are polynomials where \(r\ne 0,\) then
\(\ \frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}\) and \(\frac{p}{r}-\frac{q}{r}=\frac{p-q}{r}\)
How to find the least common denominator of rational expressions.
1. Factor each expression completely.
2. List the factors of each expression. Match factors vertically when possible.
3. Bring down the columns.
4. Write the LCD as the product of the factors.
How to add or subtract rational expressions.
1. Determine if the expressions have a common denominator.
  - Yes – go to step 2.
  - No – Rewrite each rational expression with the LCD.
    - Find the LCD.
    - Rewrite each rational expression as an equivalent rational expression with the LCD.
2. Add or subtract the rational expressions.
3. Simplify, if possible.

Add and Subtract Rational Expressions

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, add.

Try it.

\(\frac{2}{15}+\frac{7}{15}\)

Solution

\(\frac{3}{5}\)

Try it.

\(\frac{7}{24}+\frac{11}{24}\)

Try it.

\(\frac{3c}{4c-5}+\frac{5}{4c-5}\)

Solution

\(\frac{3c+5}{4c-5}\)

Try it.

\(\frac{7m}{2m+n}+\frac{4}{2m+n}\)

Try it.

\(\frac{2{r}^{2}}{2r-1}+\frac{15r-8}{2r-1}\)

Solution

\(r+8\)

Try it.

\(\frac{3{s}^{2}}{3s-2}+\frac{13s-10}{3s-2}\)

Try it.

\(\frac{2{w}^{2}}{{w}^{2}-16}+\frac{8w}{{w}^{2}-16}\)

Solution

\(\frac{2w}{w-4}\)

Try it.

\(\frac{7{x}^{2}}{{x}^{2}-9}+\frac{21x}{{x}^{2}-9}\)

In the following exercises, subtract.

Try it.

\(\frac{9{a}^{2}}{3a-7}-\frac{49}{3a-7}\)

Solution

\(3a+7\)

Try it.

\(\frac{25{b}^{2}}{5b-6}-\frac{36}{5b-6}\)

Try it.

\(\frac{3{m}^{2}}{6m-30}-\frac{21m-30}{6m-30}\)

Solution

\(\frac{m-2}{2}\)

Try it.

\(\frac{2{n}^{2}}{4n-32}-\frac{18n-16}{4n-32}\)

Try it.

\(\frac{6{p}^{2}+3p+4}{{p}^{2}+4p-5}-\frac{5{p}^{2}+p+7}{{p}^{2}+4p-5}\)

Solution

\(\frac{p+3}{p+5}\)

Try it.

\(\frac{5{q}^{2}+3q-9}{{q}^{2}+6q+8}-\frac{4{q}^{2}+9q+7}{{q}^{2}+6q+8}\)

Try it.

\(\frac{5{r}^{2}+7r-33}{{r}^{2}-49}-\frac{4{r}^{2}+5r+30}{{r}^{2}-49}\)

Solution

\(\frac{r+9}{r+7}\)

Try it.

\(\frac{7{t}^{2}-t-4}{{t}^{2}-25}-\frac{6{t}^{2}+12t-44}{{t}^{2}-25}\)

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add or subtract.

Try it.

\(\frac{10v}{2v-1}+\frac{2v+4}{1-2v}\)

Solution

\(4\)

Try it.

\(\frac{20w}{5w-2}+\frac{5w+6}{2-5w}\)

Try it.

\(\frac{10{x}^{2}+16x-7}{8x-3}+\frac{2{x}^{2}+3x-1}{3-8x}\)

Solution

\(x+2\)

Try it.

\(\frac{6{y}^{2}+2y-11}{3y-7}+\frac{3{y}^{2}-3y+17}{7-3y}\)

Try it.

\(\frac{{z}^{2}+6z}{{z}^{2}-25}-\frac{3z+20}{25-{z}^{2}}\)

Solution

\(\frac{z+4}{z-5}\)

Try it.

\(\frac{{a}^{2}+3a}{{a}^{2}-9}-\frac{3a-27}{9-{a}^{2}}\)

Try it.

\(\frac{2{b}^{2}+30b-13}{{b}^{2}-49}-\frac{2{b}^{2}-5b-8}{49-{b}^{2}}\)

Solution

\(\frac{4b-3}{b-7}\)

Try it.

\(\frac{{c}^{2}+5c-10}{{c}^{2}-16}-\frac{{c}^{2}-8c-10}{16-{c}^{2}}\)

Find the Least Common Denominator of Rational Expressions

In the following exercises, ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

Try it.

\(\frac{5}{{x}^{2}-2x-8},\frac{2x}{{x}^{2}-x-12}\)

Solution

ⓐ \((x+2)(x-4)(x+3)\)
ⓑ \(\frac{5x+15}{(x+2)(x-4)(x+3)}\),
\(\frac{2{x}^{2}+4x}{(x+2)(x-4)(x+3)}\)

Try it.

\(\frac{8}{{y}^{2}+12y+35},\frac{3y}{{y}^{2}+y-42}\)

Try it.

\(\frac{9}{{z}^{2}+2z-8},\frac{4z}{{z}^{2}-4}\)

Solution

ⓐ \((z-2)(z+4)(z+2)\)
ⓑ \(\frac{9z+18}{(z-2)(z+4)(z+2)}\),
\(\frac{4{z}^{2}+16z}{(z-2)(z+4)(z+2)}\)

Try it.

\(\frac{6}{{a}^{2}+14a+45},\frac{5a}{{a}^{2}-81}\)

Try it.

\(\frac{4}{{b}^{2}+6b+9},\frac{2b}{{b}^{2}-2b-15}\)

Solution

ⓐ \((b+3)(b+3)(b-5)\)
ⓑ \(\frac{4b-20}{(b+3)(b+3)(b-5)}\),
\(\frac{2{b}^{2}+6b}{(b+3)(b+3)(b-5)}\)

Try it.

\(\frac{5}{{c}^{2}-4c+4},\frac{3c}{{c}^{2}-7c+10}\)

Try it.

\(\frac{2}{3{d}^{2}+14d-5},\frac{5d}{3{d}^{2}-19d+6}\)

Solution

ⓐ \((d+5)(3d-1)(d-6)\)
ⓑ \(\frac{2d-12}{(d+5)(3d-1)(d-6)}\),
\(\frac{5{d}^{2}+25d}{(d+5)(3d-1)(d-6)}\)

Try it.

\(\frac{3}{5{m}^{2}-3m-2},\frac{6m}{5{m}^{2}+17m+6}\)

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

Try it.

\(\frac{7}{10{x}^{2}y}+\frac{4}{15x{y}^{2}}\)

Solution

\(\frac{21y+8x}{30{x}^{2}{y}^{2}}\)

Try it.

\(\frac{1}{12{a}^{3}{b}^{2}}+\frac{5}{9{a}^{2}{b}^{3}}\)

Try it.

\(\frac{3}{r+4}+\frac{2}{r-5}\)

Solution

\(\frac{5r-7}{(r+4)(r-5)}\)

Try it.

\(\frac{4}{s-7}+\frac{5}{s+3}\)

Try it.

\(\frac{5}{3w-2}+\frac{2}{w+1}\)

Solution

\(\frac{11w+1}{(3w-2)(w+1)}\)

Try it.

\(\frac{4}{2x+5}+\frac{2}{x-1}\)

Try it.

\(\frac{2y}{y+3}+\frac{3}{y-1}\)

Solution

\(\frac{2{y}^{2}+y+9}{(y+3)(y-1)}\)

Try it.

\(\frac{3z}{z-2}+\frac{1}{z+5}\)

Try it.

\(\frac{5b}{{a}^{2}b-2{a}^{2}}+\frac{2b}{{b}^{2}-4}\)

Solution

\(\frac{b(5b+10+2{a}^{2})}{{a}^{2}(b-2)(b+2)}\)

Try it.

\(\frac{4}{cd+3c}+\frac{1}{{d}^{2}-9}\)

Try it.

\(\frac{-3m}{3m-3}+\frac{5m}{{m}^{2}+3m-4}\)

Solution

\(\text{-}\frac{m}{m+4}\)

Try it.

\(\frac{8}{4n+4}+\frac{6}{{n}^{2}-n-2}\)

Try it.

\(\frac{3r}{{r}^{2}+7r+6}+\frac{9}{{r}^{2}+4r+3}\)

Solution

\(\frac{3({r}^{2}+6r+18)}{(r+1)(r+6)(r+3)}\)

Try it.

\(\frac{2s}{{s}^{2}+2s-8}+\frac{4}{{s}^{2}+3s-10}\)

Try it.

\(\frac{t}{t-6}-\frac{t-2}{t+6}\)

Solution

\(\frac{2(7t-6)}{(t-6)(t+6)}\)

Try it.

\(\frac{x-3}{x+6}-\frac{x}{x+3}\)

Try it.

\(\frac{5a}{a+3}-\frac{a+2}{a+6}\)

Solution

\(\frac{4{a}^{2}+25a-6}{(a+3)(a+6)}\)

Try it.

\(\frac{3b}{b-2}-\frac{b-6}{b-8}\)

Try it.

\(\frac{6}{m+6}-\frac{12m}{{m}^{2}-36}\)

Solution

\(\frac{-6}{m-6}\)

Try it.

\(\frac{4}{n+4}-\frac{8n}{{n}^{2}-16}\)

Try it.

\(\frac{-9p-17}{{p}^{2}-4p-21}-\frac{p+1}{7-p}\)

Solution

\(\frac{p+2}{p+3}\)

Try it.

\(\frac{-13q-8}{{q}^{2}+2q-24}-\frac{q+2}{4-q}\)

Try it.

\(\frac{-2r-16}{{r}^{2}+6r-16}-\frac{5}{2-r}\)

Solution

\(\frac{3}{r-2}\)

Try it.

\(\frac{2t-30}{{t}^{2}+6t-27}-\frac{2}{3-t}\)

Try it.

\(\frac{2x+7}{10x-1}+3\)

Solution

\(\frac{4(8x+1)}{10x-1}\)

Try it.

\(\frac{8y-4}{5y+2}-6\)

Try it.

\(\frac{3}{{x}^{2}-3x-4}-\frac{2}{{x}^{2}-5x+4}\)

Solution

\(\frac{x-5}{(x-4)(x+1)(x-1)}\)

Try it.

\(\frac{4}{{x}^{2}-6x+5}-\frac{3}{{x}^{2}-7x+10}\)

Try it.

\(\frac{5}{{x}^{2}+8x-9}-\frac{4}{{x}^{2}+10x+9}\)

Solution

\(\frac{1}{(x-1)(x+1)}\)

Try it.

\(\frac{3}{2{x}^{2}+5x+2}-\frac{1}{2{x}^{2}+3x+1}\)

Try it.

\(\frac{5a}{a-2}+\frac{9}{a}-\frac{2a+18}{{a}^{2}-2a}\)

Solution

\(\frac{5{a}^{2}+7a-36}{a(a-2)}\)

Try it.

\(\frac{2b}{b-5}+\frac{3}{2b}-\frac{2b-15}{2{b}^{2}-10b}\)

Try it.

\(\frac{c}{c+2}+\frac{5}{c-2}-\frac{10c}{{c}^{2}-4}\)

Solution

\(\frac{c-5}{c+2}\)

Try it.

\(\frac{6d}{d-5}+\frac{1}{d+4}-\frac{7d-5}{{d}^{2}-d-20}\)

Try it.

\(\frac{3d}{d+2}+\frac{4}{d}-\frac{d+8}{{d}^{2}+2d}\)

Solution

\(\frac{3(d+1)}{d+2}\)

Try it.

\(\frac{2q}{q+5}+\frac{3}{q-3}-\frac{13q+15}{{q}^{2}+2q-15}\)

Add and Subtract Rational Functions

In the following exercises, find ⓐ \(R(x)=f(x)+g(x)\) ⓑ \(R(x)=f(x)-g(x).\)

Try it.

\(f(x)=\frac{-5x-5}{{x}^{2}+x-6}\) and
\(\ g(x)=\frac{x+1}{2-x}\)

Solution

ⓐ \(R(x)=-\frac{(x+8)(x+1)}{(x-2)(x+3)}\) ⓑ \(R(x)=\frac{x+1}{x+3}\)

Try it.

\(f(x)=\frac{-4x-24}{{x}^{2}+x-30}\) and
\(\ g(x)=\frac{x+7}{5-x}\)

Try it.

\(f(x)=\frac{6x}{{x}^{2}-64}\) and
\(\ g(x)=\frac{3}{x-8}\)

Solution

ⓐ \(\frac{3(3x+8)}{(x-8)(x+8)}\)
ⓑ \(R(x)=\frac{3}{x+8}\)

Try it.

\(f(x)=\frac{5}{x+7}\) and
\(\ g(x)=\frac{10x}{{x}^{2}-49}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Add: \(\frac{7}{10}+\frac{8}{15}.\)
If you missed this problem, review .

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\(\frac{37}{30}\)
Subtract: \(\frac{3x}{4}-\frac{8}{9}.\)
If you missed this problem, review .

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\(\frac{27x-32}{36}\)
Subtract: \(6(2x+1)-4(x-5).\)
If you missed this problem, review .

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\(8x+26\)

Add: \(\frac{11x+28}{x+4}+\frac{{x}^{2}}{x+4}.\)

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Since the denominator is \(x+4,\) we must exclude the value \(x=-4.\)

	\(\ \frac{11x+28}{x+4}+\frac{{x}^{2}}{x+4},\ x\ne \text{-}4\)
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\ \frac{11x+28+{x}^{2}}{x+4}\)
Write the degrees in descending order.	\(\ \frac{{x}^{2}+11x+28}{x+4}\)
Factor the numerator.	\(\ \frac{(x+4)(x+7)}{x+4}\)
Simplify by removing common factors.	\(\ \frac{(x+4)(x+7)}{x+4}\)
Simplify.	\(\ x+7\)

The expression simplifies to \(x+7\) but the original expression had a denominator of \(x+4\) so \(x\ne \text{-}4.\)

Simplify: \(\frac{9x+14}{x+7}+\frac{{x}^{2}}{x+7}.\)

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\(x+2\)
Simplify: \(\frac{{x}^{2}+8x}{x+5}+\frac{15}{x+5}.\)

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\(x+3\)

Subtract: \(\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}.\)

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	\(\ \frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}\)
Subtract the numerators and place the difference over the common denominator.	\(\ \frac{5{x}^{2}-7x+3-(4{x}^{2}+x-9)}{{x}^{2}-3x-18}\)
Distribute the sign in the numerator.	\(\ \frac{5{x}^{2}-7x+3-4{x}^{2}-x+9}{{x}^{2}-3x-18}\)
Combine like terms.	\(\ \frac{{x}^{2}-8x+12}{{x}^{2}-3x-18}\)
Factor the numerator and the denominator.	\(\ \frac{(x-2)(x-6)}{(x+3)(x-6)}\)
Simplify by removing common factors.	\(\ \frac{(x-2)(x-6)}{(x+3)(x-6)}\)
	\(\ \frac{(x-2)}{(x+3)}\)

Subtract: \(\frac{4{x}^{2}-11x+8}{{x}^{2}-3x+2}-\frac{3{x}^{2}+x-3}{{x}^{2}-3x+2}.\)

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\(\frac{x-11}{x-2}\)
Subtract: \(\frac{6{x}^{2}-x+20}{{x}^{2}-81}-\frac{5{x}^{2}+11x-7}{{x}^{2}-81}.\)

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\(\frac{x-3}{x+9}\)

Subtract: \(\frac{{m}^{2}-6m}{{m}^{2}-1}-\frac{3m+2}{1-{m}^{2}}.\)

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The denominators are opposites, so multiply the second fraction by \(\frac{-1}{-1}.\)
Simplify the second fraction.
The denominators are the same. Subtract the numerators.
Distribute.
Combine like terms.
Factor the numerator and denominator.
Simplify by removing common factors.
Simplify.

Subtract: \(\frac{{y}^{2}-5y}{{y}^{2}-4}-\frac{6y-6}{4-{y}^{2}}.\)

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\(\frac{y+3}{y+2}\)
Subtract: \(\frac{2{n}^{2}+8n-1}{{n}^{2}-1}-\frac{{n}^{2}-7n-1}{1-{n}^{2}}.\)

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\(\frac{3n-2}{n-1}\)
ⓐ Find the LCD for the expressions \(\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}\) and ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

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ⓐ
Find the LCD for \(\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}.\)
Factor each denominator completely, lining up common factors.

Bring down the columns.
Write the LCD as the product of the factors.
ⓑ
Factor each denominator.
Multiply each denominator by the ‘missing’
LCD factor and multiply each numerator by the same factor.
Simplify the numerators.
ⓐ Find the LCD for the expressions \(\frac{2}{{x}^{2}-x-12},\frac{1}{{x}^{2}-16}\) ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

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ⓐ \((x-4)(x+3)(x+4)\)
ⓑ \(\frac{2x+8}{(x-4)(x+3)(x+4)}\),
\(\frac{x+3}{(x-4)(x+3)(x+4)}\)
ⓐ Find the LCD for the expressions \(\frac{3x}{{x}^{2}-3x-10},\frac{5}{{x}^{2}+3x+2}\) ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

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ⓐ \((x+2)(x-5)(x+1)\)
ⓑ \(\frac{3{x}^{2}+3x}{(x+2)(x-5)(x+1)}\),
\(\frac{5x-25}{(x+2)(x-5)(x+1)}\)
Add: \(\frac{3}{x-3}+\frac{2}{x-2}.\)
Add: \(\frac{2}{x-2}+\frac{5}{x+3}.\)

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\(\frac{7x-4}{(x-2)(x+3)}\)
Add:\(\frac{4}{m+3}+\frac{3}{m+4}.\)

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\(\frac{7m+25}{(m+3)(m+4)}\)

Add: \(\frac{8}{{x}^{2}-2x-3}+\frac{3x}{{x}^{2}+4x+3}.\)

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Do the expressions have a common denominator?	No.
Rewrite each expression with the LCD.
\(\begin{array}{llll} \\ \\ \text{Find the LCD.} & & & \begin{array}{l}\ {x}^{2}-2x-3=(x+1)(x-3) \\ \underset{______________________________}{{x}^{2}+4x+3=(x+1)\ (x+3)}\ \\ \\ \text{LCD}\ =(x+1)(x-3)(x+3)\end{array}\end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Add the rational expressions.
Simplify the numerator.
	The numerator is prime, so there are no common factors.

Add: \(\frac{1}{{m}^{2}-m-2}+\frac{5m}{{m}^{2}+3m+2}.\)

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\(\frac{5{m}^{2}-9m+2}{(m+1)(m-2)(m+2)}\)
Add:\(\frac{2n}{{n}^{2}-3n-10}+\frac{6}{{n}^{2}+5n+6}.\)

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\(\frac{2{n}^{2}+12n-30}{(n+2)(n-5)(n+3)}\)

Subtract: \(\frac{8y}{{y}^{2}-16}-\frac{4}{y-4}.\)

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Do the expressions have a common denominator?	No.
Rewrite each expression with the LCD.
\(\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}{y}^{2}-16=(y-4)(y+4) \\ \underset{____________}{\text{}y-4\ =y-4} \\ \text{LCD}\ =(y-4)(y+4)\end{array}\end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Subtract the rational expressions.
Simplify the numerator.
Factor the numerator to look for common factors.
Remove common factors
Simplify.

Subtract: \(\frac{2x}{{x}^{2}-4}-\frac{1}{x+2}.\)

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\(\frac{1}{x-2}\)
Subtract: \(\frac{3}{z+3}-\frac{6z}{{z}^{2}-9}.\)

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\(\frac{-3}{z-3}\)

Subtract:\(\frac{-3n-9}{{n}^{2}+n-6}-\frac{n+3}{2-n}.\)

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Factor the denominator.
Since \(n-2\) and \(2-n\) are opposites, we will multiply the second rational expression by \(\frac{-1}{-1}.\)

Simplify. Remember, \(a-(\text{-}b)=a+b.\)
Do the rational expressions have a common denominator? No.
\(\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}{n}^{2}+n-6=(n-2)(n+3) \\ \underset{_________________}{n-2=(n-2)} \\ \text{LCD}\ =(n-2)(n+3)\end{array}\end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Add the rational expressions.
Simplify the numerator.
Factor the numerator to look for common factors.
Simplify.

Subtract :\(\frac{3x-1}{{x}^{2}-5x-6}-\frac{2}{6-x}.\)

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\(\frac{5x+1}{(x-6)(x+1)}\)
Subtract: \(\frac{-2y-2}{{y}^{2}+2y-8}-\frac{y-1}{2-y}.\)

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\(\frac{y+3}{y+4}\)

Subtract: \(\frac{4}{{a}^{2}+6a+5}-\frac{3}{{a}^{2}+7a+10}.\)

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Factor the denominators.
Do the rational expressions have a common denominator? No.
\(\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}\ {a}^{2}+6a+5\ =(a+1)(a+5) \\ \underset{____________________________}{{a}^{2}+7a+10\ =\ (a+5)(a+2)} \\ \text{LCD}=(a+1)(a+5)(a+2)\end{array}\end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Subtract the rational expressions.
Simplify the numerator.

Look for common factors.
Simplify.

Subtract: \(\frac{3}{{b}^{2}-4b-5}-\frac{2}{{b}^{2}-6b+5}.\)

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\(\frac{1}{(b+1)(b-1)}\)
Subtract: \(\frac{4}{{x}^{2}-4}-\frac{3}{{x}^{2}-x-2}.\)

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\(\frac{1}{(x+2)(x+1)}\)

Simplify: \(\frac{2u}{u-1}+\frac{1}{u}-\frac{2u-1}{{u}^{2}-u}.\)

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Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
\(\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}u-1\ =\ (u-1) \\ u\ =u \\ \underset{_______________}{{u}^{2}-u=u(u-1)} \\ \text{LCD}\ =u(u-1)\end{array}\end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.

Write as one rational expression.
Simplify.
Factor the numerator, and remove common factors.
Simplify.

Simplify: \(\frac{v}{v+1}+\frac{3}{v-1}-\frac{6}{{v}^{2}-1}.\)

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\(\frac{v+3}{v+1}\)
Simplify: \(\frac{3w}{w+2}+\frac{2}{w+7}-\frac{17w+4}{{w}^{2}+9w+14}.\)

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\(\frac{3w}{w+7}\)

Find \(R(x)=f(x)-g(x)\) where \(f(x)=\frac{x+5}{x-2}\) and \(g(x)=\frac{5x+18}{{x}^{2}-4}.\)

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Substitute in the functions \(f(x),\)\(g(x).\)
Factor the denominators.
Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
\(\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}x-2\ =(x-2) \\ \underset{___________________}{{x}^{2}-4=(x-2)(x+2)} \\ \text{LCD}=(x-2)(x+2)\end{array}\end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Write as one rational expression.
Simplify.

Factor the numerator, and remove common factors.
Simplify.

Find \(R(x)=f(x)-g(x)\) where \(f(x)=\frac{x+1}{x+3}\) and \(g(x)=\frac{x+17}{{x}^{2}-x-12}.\)

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\(\frac{x-7}{x-4}\)
Find \(R(x)=f(x)+g(x)\) where \(f(x)=\frac{x-4}{x+3}\) and \(g(x)=\frac{4x+6}{{x}^{2}-9}.\)

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\(\frac{{x}^{2}-3x+18}{(x+3)(x-3)}\)
\(\frac{2}{15}+\frac{7}{15}\)

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\(\frac{3}{5}\)
\(\frac{7}{24}+\frac{11}{24}\)
\(\frac{3c}{4c-5}+\frac{5}{4c-5}\)

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\(\frac{3c+5}{4c-5}\)
\(\frac{7m}{2m+n}+\frac{4}{2m+n}\)

Symbols used here

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Add and Subtract Rational Expressions

Add and subtract rational expressions with a common denominator
Add and subtract rational expressions whose denominators are opposites
Find the least common denominator of rational expressions
Add and subtract rational expressions with unlike denominators
Add and subtract rational functions
Factor each denominator completely.
List the factors of each denominator. Match factors vertically when possible.
Bring down the columns by including all factors, but do not include common factors twice.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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