maths.free › Algebra › 8. Rational Expressions and Equations › Add and Subtract Rational Expressions with Unlike Denominators

Add and Subtract Rational Expressions with Unlike Denominators

Find the least common denominator of rational expressions

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 the example \(\frac{7}{12}+\frac{5}{18}\) from Foundations. Since the denominators are not the same, the first step was to find the least common denominator (LCD). Remember, the LCD is the least common multiple of the denominators. It is the smallest number we can use as a common denominator.

To find the LCD of 12 and 18, we factored each number 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.

\[\frac{\begin{array}{l} \\ \\ 12=2\cdot 2\cdot 3 \\ 18=2\cdot 3\cdot 3\end{array}}{\begin{array}{l} \\ \\ \text{LCD}=2\cdot 2\cdot 3\cdot 3 \\ \text{LCD}=36\end{array}}\]

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

Example

Try it.

Find the LCD for \(\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}\).

Solution

	\(\text{Find the LCD for}\ \frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}.\)
Factor each expression completely, lining up common factors. Bring down the columns.	\(\frac{\begin{array}{l}{x}^{2}-2x-3=(x+1)(x-3) \\ {x}^{2}+4x+3=(x+1)\ (x+3)\end{array}}{\ \text{LCD}=(x+1)(x-3)(x+3)}\)
Multiply the factors.	\(\text{The LCD is}\ (x+1)(x-3)(x+3).\)

Find Equivalent Rational Expressions

When we add numerical fractions, once we find the LCD, we rewrite each fraction as an equivalent fraction with the LCD.

We will do the same thing for rational expressions.

Example

Try it.

Rewrite as equivalent rational expressions with denominator \((x+1)(x-3)(x+3)\): \(\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}.\)

Solution


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

Add Rational Expressions with Different Denominators

Now we have all the steps we need to add rational expressions with different denominators. As we have done previously, we will do one example of adding numerical fractions first.

Example

Try it.

Add: \(\frac{7}{12}+\frac{5}{18}.\)

Solution


Find the LCD of 12 and 18.
Rewrite each fraction as an equivalent fraction with the LCD.
Add the fractions.
The fraction cannot be simplified.

Now we will add rational expressions whose denominators are monomials.

Example

Try it.

Add: \(\frac{5}{12{x}^{2}y}+\frac{4}{21x{y}^{2}}.\)

Solution


Find the LCD of \(12{x}^{2}y\) and \(21x{y}^{2}\).

Rewrite each rational expression as an equivalent fraction with the LCD.
Simplify.
Add the rational expressions.
There are no factors common to the numerator and denominator. The fraction cannot be simplified.

Now we are ready to tackle polynomial denominators.

How to Add Rational Expressions with Different Denominators

Try it.

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

Solution

The steps to use to add rational expressions are summarized in the following procedure box.

Example

Try it.

Add: \(\frac{2a}{2ab+{b}^{2}}+\frac{3a}{4{a}^{2}-{b}^{2}}.\)

Solution


Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
Find the LCD.
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.
There are no factors common to the numerator and denominator. The fraction cannot be simplified.

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

Subtract Rational Expressions with Different Denominators

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.

How to Subtract Rational Expressions with Different Denominators

Try it.

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

Solution

The steps to take to subtract rational expressions are listed below.

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.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Subtract the rational expressions.
Simplify the numerators.
Factor the numerator to look for common factors.
Remove common factors.
Simplify.

There are lots of negative signs in the next example. Be extra careful!

Example

Try it.

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

Solution


Factor the denominator.
Since \(n-2\) and \(2-n\) are opposites, we will mutliply the second rational expression by\(\frac{-1}{-1}\).
Simplify.
Do the expressions have a common denominator? No.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Simplify the rational expressions.
Simplify the numerator.
Factor the numerator to look for common factors.
Simplify.

When one expression is not in fraction form, we can write it as a fraction with denominator 1.

We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example we will start by factoring all three denominators to find their LCD.

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

Key Concepts

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. Multiply the factors.
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 with Unlike Denominators

In the following exercises, find the LCD.

Try it.

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

Solution

\((x-4)(x+2)(x+3)\)

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

\((z-2)(z+4)(z+2)\)

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

\((b+3)(b+3)(b-5)\)

Try it.

\(\frac{5}{{c}^{2}-4c+4},\frac{3c}{{c}^{2}-10c+16}\)

Try it.

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

Solution

\((3d-1)(d+5)(d-6)\)

Try it.

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

In the following exercises, write as equivalent rational expressions with the given LCD.

Try it.

\(\frac{5}{{x}^{2}-2x-8},\frac{2x}{{x}^{2}-x-12}\)
LCD \((x-4)(x+2)(x+3)\)

Solution

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

Try it.

\(\frac{8}{{y}^{2}+12y+35},\frac{3y}{{y}^{2}+y-42}\)
LCD \((y+7)(y+5)(y-6)\)

Try it.

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

Solution

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

Try it.

\(\frac{6}{{a}^{2}+14a+45},\frac{5a}{{a}^{2}-81}\)
LCD \((a+9)(a+5)(a-9)\)

Try it.

\(\frac{4}{{b}^{2}+6b+9},\frac{2b}{{b}^{2}-2b-15}\)
LCD \((b+3)(b+3)(b-5)\)

Solution

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

Try it.

\(\frac{5}{{c}^{2}-4c+4},\frac{3c}{{c}^{2}-10c+16}\)
LCD \((c-2)(c-2)(c-8)\)

Try it.

\(\frac{2}{3{d}^{2}+14d-5},\frac{5d}{3{d}^{2}-19d+6}\)
LCD \((3d-1)(d+5)(d-6)\)

Solution

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

Try it.

\(\frac{3}{5{m}^{2}-3m-2},\frac{6m}{5{m}^{2}+17m+6}\)
LCD \((5m+2)(m-1)(m+3)\)

In the following exercises, add.

Try it.

\(\frac{5}{24}+\frac{11}{36}\)

Solution

\(\frac{37}{72}\)

Try it.

\(\frac{7}{30}+\frac{13}{45}\)

Try it.

\(\frac{9}{20}+\frac{11}{30}\)

Solution

\(\frac{49}{60}\)

Try it.

\(\frac{8}{27}+\frac{7}{18}\)

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.

\(\frac{1}{2m}+\frac{7}{8{m}^{2}n}\)

Solution

\(\frac{4mn+7}{8{m}^{2}n}\)

Try it.

\(\frac{5}{6{p}^{2}q}+\frac{1}{4p}\)

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.

\(\frac{8}{t+5}+\frac{6}{t-5}\)

Solution

\(\frac{14t-10}{(t+5)(t-5)}\)

Try it.

\(\frac{7}{v+5}+\frac{9}{v-5}\)

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.

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

Solution

\(\frac{2{m}^{2}+23m}{3(m-1)(m+4)}\)

Try it.

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

Try it.

\(\frac{3}{{n}^{2}+3n-18}+\frac{4n}{{n}^{2}+8n+12}\)

Solution

\(\frac{4{n}^{2}-9n+6}{(n-3)(n+6)(n+2)}\)

Try it.

\(\frac{6}{{q}^{2}-3q-10}+\frac{5q}{{q}^{2}-8q+15}\)

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.

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

In the following exercises, subtract.

Try it.

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

Solution

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

Try it.

\(\frac{v}{v-3}-\frac{v-6}{v+1}\)

Try it.

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

Solution

\(\frac{-4(1+w)}{(w+4)(w-2)}\)

Try it.

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

Try it.

\(\frac{y-4}{y+1}-\frac{1}{y+7}\)

Solution

\(\frac{{y}^{2}+2y-29}{(y+1)(y+7)}\)

Try it.

\(\frac{z+8}{z-3}-\frac{z}{z-2}\)

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.

\(\frac{6c}{{c}^{2}-25}-\frac{3}{c+5}\)

Solution

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

Try it.

\(\frac{4d}{{d}^{2}-81}-\frac{2}{d+9}\)

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.

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

Solution

\(\frac{v-14}{v+3}\)

Try it.

\(\frac{6w+5}{w-1}+2\)

Try it.

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

Solution

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

Try it.

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

In the following exercises, add and subtract.

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.

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

In the following exercises, simplify.

Try it.

\(\frac{6a}{3ab+{b}^{2}}+\frac{3a}{9{a}^{2}-{b}^{2}}\)

Solution

\(\frac{3a(6a-b)}{b(3a+b)(3a-b)}\)

Try it.

\(\frac{2c}{2c+10}+\frac{7c}{{c}^{2}+9c+20}\)

Try it.

\(\frac{6d}{{d}^{2}-64}-\frac{3}{d-8}\)

Solution

\(\frac{3}{d+8}\)

Try it.

\(\frac{5}{n+7}-\frac{10n}{{n}^{2}-49}\)

Try it.

\(\frac{4m}{{m}^{2}+6m-7}+\frac{2}{{m}^{2}+10m+21}\)

Solution

\(\frac{2(2{m}^{2}+7m-1)}{(m+7)(m-1)(m+3)}\)

Try it.

\(\frac{3p}{{p}^{2}+4p-12}+\frac{1}{{p}^{2}+p-30}\)

Try it.

\(\frac{-5n-5}{{n}^{2}+n-6}+\frac{n+1}{2-n}\)

Solution

\(-\frac{\left(n+1\right)\left(n+8\right)}{\left(n+3\right)\left(n-2\right)}\)

Try it.

\(\frac{-4b-24}{{b}^{2}+b-30}+\frac{b+7}{5-b}\)

Try it.

\(\frac{7}{15p}+\frac{5}{18pq}\)

Solution

\(\frac{42q+25}{90pq}\)

Try it.

\(\frac{3}{20{a}^{2}}+\frac{11}{12a{b}^{2}}\)

Try it.

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

Solution

\(\frac{7(x+2)}{(x-2)(x+5)}\)

Try it.

\(\frac{6}{m+4}+\frac{9}{m-8}\)

Try it.

\(\frac{2q+7}{q+4}-2\)

Solution

\(\frac{-1}{q+4}\)

Try it.

\(\frac{3y-1}{y+4}-2\)

Try it.

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

Solution

\(\frac{2\left(4z+1\right)}{(z-5)(z+1)}\)

Try it.

\(\frac{t}{t-5}-\frac{t-1}{t+5}\)

Try it.

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

Solution

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

Try it.

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

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

Cavabı göstər

\(\frac{37}{30}\)
Subtract: \(6(2x+1)-4(x-5).\)
If you missed this problem, review .

Cavabı göstər

\(8x+26\)
Find the Greatest Common Factor of \(9{x}^{2}{y}^{3}\) and \(12x{y}^{5}\).
If you missed this problem, review .

Cavabı göstər

\(3x{y}^{3}\)
Factor completely \(-48n-12\).
If you missed this problem, review .

Cavabı göstər

\(-12\left(4n+1\right)\)

Find the LCD for \(\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}\).

Cavabı göstər

	\(\text{Find the LCD for}\ \frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}.\)
Factor each expression completely, lining up common factors. Bring down the columns.	\(\frac{\begin{array}{l}{x}^{2}-2x-3=(x+1)(x-3) \\ {x}^{2}+4x+3=(x+1)\ (x+3)\end{array}}{\ \text{LCD}=(x+1)(x-3)(x+3)}\)
Multiply the factors.	\(\text{The LCD is}\ (x+1)(x-3)(x+3).\)

Find the LCD for \(\frac{2}{{x}^{2}-x-12},\frac{1}{{x}^{2}-16}\).

Cavabı göstər

\((x-4)(x+4)(x+3)\)
Find the LCD for \(\frac{x}{{x}^{2}+8x+15},\frac{5}{{x}^{2}+9x+18}\).

Cavabı göstər

\((x+3)(x+6)(x+5)\)
Rewrite as equivalent rational expressions with denominator \((x+1)(x-3)(x+3)\): \(\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}.\)

Cavabı göstər

Factor each denominator.
Find the LCD.
Multiply each denominator by the 'missing' factor and multiply each numerator by the same factor.
Simplify the numerators.
Rewrite as equivalent rational expressions with denominator \((x+3)(x-4)(x+4)\):
\(\frac{2}{{x}^{2}-x-12},\frac{1}{{x}^{2}-16}.\)

Cavabı göstər

\(\frac{2x+8}{(x-4)(x+3)(x+4)},\)
\(\frac{x+3}{(x-4)(x+3)(x+4)}\)
Rewrite as equivalent rational expressions with denominator \((x+3)(x+5)(x+6)\):
\(\frac{x}{{x}^{2}+8x+15},\frac{5}{{x}^{2}+9x+18}.\)

Cavabı göstər

\(\frac{{x}^{2}+6x}{(x+3)(x+5)(x+6)},\)
\(\frac{5x+25}{(x+3)(x+5)(x+6)}\)
Add: \(\frac{7}{12}+\frac{5}{18}.\)

Cavabı göstər

Find the LCD of 12 and 18.
Rewrite each fraction as an equivalent fraction with the LCD.
Add the fractions.
The fraction cannot be simplified.
Add: \(\frac{11}{30}+\frac{7}{12}.\)

Cavabı göstər

\(\frac{19}{20}\)
Add: \(\frac{3}{8}+\frac{9}{20}.\)

Cavabı göstər

\(\frac{33}{40}\)

Add: \(\frac{5}{12{x}^{2}y}+\frac{4}{21x{y}^{2}}.\)

Cavabı göstər


Find the LCD of \(12{x}^{2}y\) and \(21x{y}^{2}\).

Rewrite each rational expression as an equivalent fraction with the LCD.
Simplify.
Add the rational expressions.
There are no factors common to the numerator and denominator. The fraction cannot be simplified.

Add: \(\frac{2}{15{a}^{2}b}+\frac{5}{6a{b}^{2}}.\)

Cavabı göstər

\(\frac{4b+25a}{30{a}^{2}{b}^{2}}\)
Add: \(\frac{5}{16c}+\frac{3}{8c{d}^{2}}.\)

Cavabı göstər

\(\frac{5{d}^{2}+6}{16c{d}^{2}}\)
Add: \(\frac{3}{x-3}+\frac{2}{x-2}.\)
Add: \(\frac{2}{x-2}+\frac{5}{x+3}.\)

Cavabı göstər

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

Cavabı göstər

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

Add: \(\frac{2a}{2ab+{b}^{2}}+\frac{3a}{4{a}^{2}-{b}^{2}}.\)

Cavabı göstər


Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
Find the LCD.
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.
There are no factors common to the numerator and denominator. The fraction cannot be simplified.

Add: \(\frac{5x}{xy-{y}^{2}}+\frac{2x}{{x}^{2}-{y}^{2}}.\)

Cavabı göstər

\(\frac{x(5x+7y)}{y(x-y)(x+y)}\)
Add: \(\frac{7}{2m+6}+\frac{4}{{m}^{2}+4m+3}.\)

Cavabı göstər

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

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

Cavabı göstər


Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
Find the LCD.
Rewrite each rational expression as an equivalent fraction with the LCD.
Simplify the numerators.
Add the rational expressions.
Simplify the numerator.
The numerator is prime, so there are no common factors.

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

Cavabı göstər

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

Cavabı göstər

\(\frac{2({n}^{2}+6n-15)}{(n+2)(n-5)(n+3)}\)
Subtract: \(\frac{x}{x-3}-\frac{x-2}{x+3}.\)
Subtract: \(\frac{y}{y+4}-\frac{y-2}{y-5}.\)

Cavabı göstər

\(\frac{-7y+8}{(y+4)(y-5)}\)
Subtract: \(\frac{z+3}{z+2}-\frac{z}{z+3}.\)

Cavabı göstər

\(\frac{4z+9}{(z+2)(z+3)}\)

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

Cavabı göstər


Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Subtract the rational expressions.
Simplify the numerators.
Factor the numerator to look for common factors.
Remove common factors.
Simplify.

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

Cavabı göstər

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

Cavabı göstər

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

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

Cavabı göstər


Factor the denominator.
Since \(n-2\) and \(2-n\) are opposites, we will mutliply the second rational expression by\(\frac{-1}{-1}\).
Simplify.
Do the expressions have a common denominator? No.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Simplify the rational expressions.
Simplify the numerator.
Factor the numerator to look for common factors.
Simplify.

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

Cavabı göstər

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

Cavabı göstər

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

Subtract: \(\frac{5c+4}{c-2}-3.\)

Cavabı göstər


Write \(3\) as \(\frac{3}{1}\) to have 2 rational expressions.
Do the rational expressions have a common denominator? No.
Find the LCD of \(c-2\) and \(1.\) LCD = \(c-2.\)
Rewrite \(\frac{3}{1}\) as an equivalent rational expression with the LCD.
Simplify.
Subtract the rational expressions.
Simplify.
Factor to check for common factors.
There are no common factors; the rational expression is simplified.

Subtract: \(\frac{2x+1}{x-7}-3.\)

Cavabı göstər

\(\frac{\text{-}x+22}{x-7}\)
Subtract: \(\frac{4y+3}{2y-1}-5.\)

Cavabı göstər

\(\frac{-2(3y-4)}{2y-1}\)

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

Cavabı göstər


Do the rational expressions have a common denominator? No.
Find the LCD.
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: \(\frac{v}{v+1}+\frac{3}{v-1}-\frac{6}{{v}^{2}-1}.\)

Cavabı göstər

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

Cavabı göstər

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

Symbols used here

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

derivative
Instantaneous rate of change; slope of the graph.

\leq,\ \geq

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

\pm

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

\neq

not equal
The two sides are different.

\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 Unlike Denominators

Find the least common denominator of rational expressions
Find equivalent rational expressions
Add rational expressions with different denominators
Subtract rational expressions with different denominators
Factor each expression completely.
List the factors of each expression. Match factors vertically when possible.
Bring down the columns.
Multiply the factors.

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.

Özün sına

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.

Daha çox Algebra

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