maths.free › Algebra › 8. Rational Expressions and Equations › Add and Subtract Rational Expressions with a Common Denominator

Add and Subtract Rational Expressions with a Common Denominator

Add rational expressions with a common denominator

Add 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.

We will add two numerical fractions first, to remind us of how this is done.

Example

Try it.

Add: \(\frac{5}{18}+\frac{7}{18}.\)

Solution

	\(\frac{5}{18}+\frac{7}{18}\)
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\frac{5+7}{18}\)
Add in the numerator.	\(\frac{12}{18}\)
Factor the numerator and denominator to show the common factors.	\(\frac{6\cdot 2}{6\cdot 3}\)
Remove common factors.	\(\frac{6\cdot 2}{6\cdot 3}\)
Simplify.	\(\frac{2}{3}\)

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

Example

Try it.

Add: \(\frac{3y}{4y-3}+\frac{7}{4y-3}.\)

Solution

	\(\frac{3y}{4y-3}+\frac{7}{4y-3}\)
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\frac{3y+7}{4y-3}\)

The numerator and denominator cannot be factored. The fraction is simplified.

Example

Try it.

Add: \(\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}.\)

Solution

	\(\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}\)
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\frac{7x+12+{x}^{2}}{x+3}\)
Write the degrees in descending order.	\(\frac{{x}^{2}+7x+12}{x+3}\)
Factor the numerator.	\(\frac{(x+3)(x+4)}{x+3}\)
Simplify by removing common factors.	\(\frac{(x+3)(x+4)}{x+3}\)
Simplify.	\(x+4\)

Subtract Rational Expressions with a Common Denominator

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.

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

Example

Try it.

Subtract: \(\frac{{n}^{2}}{n-10}-\frac{100}{n-10}.\)

Solution

	\(\frac{{n}^{2}}{n-10}-\frac{100}{n-10}\)
The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator.	\(\frac{{n}^{2}-100}{n-10}\)
Factor the numerator.	\(\frac{(n-10)(n+10)}{n-10}\)
Simplify by removing common factors.	\(\frac{(n-10)(n+10)}{n-10}\)
Simplify.	\(n+10\)

Be careful of the signs when you subtract a binomial!

Example

Try it.

Subtract: \(\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}.\)

Solution

	\(\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}\)
The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator.	\(\frac{{y}^{2}-(2y+24)}{y-6}\)
Distribute the sign in the numerator.	\(\frac{{y}^{2}-2y-24}{y-6}\)
Factor the numerator.	\(\frac{(y-6)(y+4)}{y-6}\)
Remove common factors.	\(\frac{(y-6)(y+4)}{y-6}\)
Simplify.	\(y+4\)

Example

Try it.

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

Solution

	\(\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}\)
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.	\(\frac{(x-2)(x-6)}{(x+3)(x-6)}\)
Simplify.	\(\frac{(x-2)}{(x+3)}\)

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.

Example

Try it.

Add: \(\frac{4u-1}{3u-1}+\frac{u}{1-3u}.\)

Solution


The denominators are opposites, so multiply the second fraction by \(\frac{-1}{-1}\).
Simplify the second fraction.
The denominators are the same. Add the numerators.
Simplify.
Simplify.

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.	\(\frac{{m}^{2}-6m+3m+2}{{m}^{2}-1}\)
Combine like terms.
Factor the numerator and denominator.
Simplify by removing common factors.
Simplify.

Key Concepts

Rational Expression Addition
- If \(p,q,\ \text{and}\ r\) are polynomials where \(r\ne 0\), then
  \[\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}\]
- To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
Rational Expression Subtraction
- If \(p,q,\ \text{and}\ r\) are polynomials where \(r\ne 0\), then
  \[\frac{p}{r}-\frac{q}{r}=\frac{p-q}{r}\]
- To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

Add and Subtract Rational Expressions with a Common Denominator

Add 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.

\(\frac{4}{21}+\frac{3}{21}\)

Try it.

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

Solution

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

Try it.

\(\frac{7}{36}+\frac{13}{36}\)

Try it.

\(\frac{3a}{a-b}+\frac{1}{a-b}\)

Solution

\(\frac{3a+1}{a-b}\)

Try it.

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

Try it.

\(\frac{d}{d+8}+\frac{5}{d+8}\)

Solution

\(\frac{d+5}{d+8}\)

Try it.

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

Try it.

\(\frac{{p}^{2}+10p}{p+2}+\frac{16}{p+2}\)

Solution

\(p+8\)

Try it.

\(\frac{{q}^{2}+12q}{q+3}+\frac{27}{q+3}\)

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.

\(\frac{8{t}^{2}}{t+4}+\frac{32t}{t+4}\)

Solution

\(8t\)

Try it.

\(\frac{6{v}^{2}}{v+5}+\frac{30v}{v+5}\)

Try it.

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

Solution

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

Try it.

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

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

Try it.

\(\frac{{y}^{2}}{y+8}-\frac{64}{y+8}\)

Solution

\(y-8\)

Try it.

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

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.

\(\frac{{c}^{2}}{c-8}-\frac{6c+16}{c-8}\)

Solution

\(c+2\)

Try it.

\(\frac{{d}^{2}}{d-9}-\frac{6d+27}{d-9}\)

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.

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.

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

In the following exercises, subtract.

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.

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

Try it.

Donald thinks that \(\frac{3}{x}+\frac{4}{x}\) is \(\frac{7}{2x}.\) Is Donald correct? Explain.

Try it.

Explain how you find the Least Common Denominator of \({x}^{2}+5x+4\) and \({x}^{2}-16.\)

Condensed — the full section is in OpenStax Elementary 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: \(\frac{y}{3}+\frac{9}{3}.\)
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

\(\frac{y+9}{3}\)
Subtract: \(\frac{10}{x}-\frac{2}{x}.\)
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

\(\frac{8}{x}\)
Factor completely: \(8{n}^{5}-20{n}^{3}.\)
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn
Factor completely: \(45{a}^{3}-5a{b}^{2}.\)
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

\(5a\left(3a-b\right)\left(3a+b\right)\)

Add: \(\frac{5}{18}+\frac{7}{18}.\)

Fi àwọn àgbèwọlé hàn

	\(\frac{5}{18}+\frac{7}{18}\)
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\frac{5+7}{18}\)
Add in the numerator.	\(\frac{12}{18}\)
Factor the numerator and denominator to show the common factors.	\(\frac{6\cdot 2}{6\cdot 3}\)
Remove common factors.	\(\frac{6\cdot 2}{6\cdot 3}\)
Simplify.	\(\frac{2}{3}\)

Add: \(\frac{7}{16}+\frac{5}{16}.\)

Fi àwọn àgbèwọlé hàn

\(\frac{3}{4}\)
Add: \(\frac{3}{10}+\frac{1}{10}.\)

Fi àwọn àgbèwọlé hàn

\(\frac{2}{5}\)
Add: \(\frac{3y}{4y-3}+\frac{7}{4y-3}.\)

Fi àwọn àgbèwọlé hàn

\(\frac{3y}{4y-3}+\frac{7}{4y-3}\)
The fractions have a common
denominator, so add the numerators and
place the sum over the common denominator. \(\frac{3y+7}{4y-3}\)

The numerator and denominator cannot be factored. The fraction is simplified.
Add: \(\frac{5x}{2x+3}+\frac{2}{2x+3}.\)

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Add: \(\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}.\)

Fi àwọn àgbèwọlé hàn

	\(\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}\)
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\frac{7x+12+{x}^{2}}{x+3}\)
Write the degrees in descending order.	\(\frac{{x}^{2}+7x+12}{x+3}\)
Factor the numerator.	\(\frac{(x+3)(x+4)}{x+3}\)
Simplify by removing common factors.	\(\frac{(x+3)(x+4)}{x+3}\)
Simplify.	\(x+4\)

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

\(x+3\)

Subtract: \(\frac{{n}^{2}}{n-10}-\frac{100}{n-10}.\)

Fi àwọn àgbèwọlé hàn

	\(\frac{{n}^{2}}{n-10}-\frac{100}{n-10}\)
The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator.	\(\frac{{n}^{2}-100}{n-10}\)
Factor the numerator.	\(\frac{(n-10)(n+10)}{n-10}\)
Simplify by removing common factors.	\(\frac{(n-10)(n+10)}{n-10}\)
Simplify.	\(n+10\)

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

Fi àwọn àgbèwọlé hàn

\(x-3\)
Subtract: \(\frac{4{x}^{2}}{2x-5}-\frac{25}{2x-5}.\)

Fi àwọn àgbèwọlé hàn

\(2x+5\)

Subtract: \(\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}.\)

Fi àwọn àgbèwọlé hàn

	\(\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}\)
The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator.	\(\frac{{y}^{2}-(2y+24)}{y-6}\)
Distribute the sign in the numerator.	\(\frac{{y}^{2}-2y-24}{y-6}\)
Factor the numerator.	\(\frac{(y-6)(y+4)}{y-6}\)
Remove common factors.	\(\frac{(y-6)(y+4)}{y-6}\)
Simplify.	\(y+4\)

Subtract: \(\frac{{n}^{2}}{n-4}-\frac{n+12}{n-4}.\)

Fi àwọn àgbèwọlé hàn

\(n+3\)
Subtract: \(\frac{{y}^{2}}{y-1}-\frac{9y-8}{y-1}.\)

Fi àwọn àgbèwọlé hàn

\(y-8\)

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

Fi àwọn àgbèwọlé hàn

	\(\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}\)
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.	\(\frac{(x-2)(x-6)}{(x+3)(x-6)}\)
Simplify.	\(\frac{(x-2)}{(x+3)}\)

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

\(\frac{x-3}{x+9}\)
Add: \(\frac{4u-1}{3u-1}+\frac{u}{1-3u}.\)

Fi àwọn àgbèwọlé hàn

The denominators are opposites, so multiply the second fraction by \(\frac{-1}{-1}\).
Simplify the second fraction.
The denominators are the same. Add the numerators.
Simplify.
Simplify.
Add: \(\frac{8x-15}{2x-5}+\frac{2x}{5-2x}.\)

Fi àwọn àgbèwọlé hàn

\(3\)
Add: \(\frac{6{y}^{2}+7y-10}{4y-7}+\frac{2{y}^{2}+2y+11}{7-4y}.\)

Fi àwọn àgbèwọlé hàn

\(y+3\)

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

Fi àwọn àgbèwọlé hàn


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.	\(\frac{{m}^{2}-6m+3m+2}{{m}^{2}-1}\)
Combine like terms.
Factor the numerator and denominator.
Simplify by removing common factors.
Simplify.

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

\(\frac{3n-2}{n-1}\)
\(\frac{2}{15}+\frac{7}{15}\)

Fi àwọn àgbèwọlé hàn

\(\frac{3}{5}\)
\(\frac{4}{21}+\frac{3}{21}\)
\(\frac{7}{24}+\frac{11}{24}\)

Fi àwọn àgbèwọlé hàn

\(\frac{3}{4}\)
\(\frac{7}{36}+\frac{13}{36}\)
\(\frac{3a}{a-b}+\frac{1}{a-b}\)

Fi àwọn àgbèwọlé hàn

\(\frac{3a+1}{a-b}\)
\(\frac{3c}{4c-5}+\frac{5}{4c-5}\)
\(\frac{d}{d+8}+\frac{5}{d+8}\)

Fi àwọn àgbèwọlé hàn

\(\frac{d+5}{d+8}\)
\(\frac{7m}{2m+n}+\frac{4}{2m+n}\)
\(\frac{{p}^{2}+10p}{p+2}+\frac{16}{p+2}\)

Fi àwọn àgbèwọlé hàn

\(p+8\)
\(\frac{{q}^{2}+12q}{q+3}+\frac{27}{q+3}\)
\(\frac{2{r}^{2}}{2r-1}+\frac{15r-8}{2r-1}\)

Fi àwọn àgbèwọlé hàn

\(r+8\)
\(\frac{3{s}^{2}}{3s-2}+\frac{13s-10}{3s-2}\)

Symbols used here

f'(x),\ \frac{dy}{dx}

derivative
Instantaneous rate of change; slope of the graph.

\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 with a Common Denominator

Add rational expressions with a common denominator
Subtract rational expressions with a common denominator
Add and subtract rational expressions whose denominators are opposites
If
To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
If
To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

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.

Wárá

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

Diẹ̀ nínú Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value