Add and Subtract Rational Expressions with Unlike Denominators

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{matrix} 12 = 2 \cdot 2 \cdot 3 \\ 18 = 2 \cdot 3 \cdot 3 \end{matrix}}{\begin{matrix} LCD = 2 \cdot 2 \cdot 3 \cdot 3 \\ LCD = 36 \end{matrix}}

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

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

Solution

Steps to find the Least Common Denominator (LCD) for given rational expressions by factoring their denominators.
	$Find the LCD for \frac{8}{x^{2} - 2 x - 3}, \frac{3 x}{x^{2} + 4 x + 3} .$
Factor each expression completely, lining up common factors. Bring down the columns.	$\frac{\begin{matrix} x^{2} - 2 x - 3 = (x + 1) (x - 3) \\ x^{2} + 4 x + 3 = (x + 1) (x + 3) \end{matrix}}{LCD = (x + 1) (x - 3) (x + 3)}$
Multiply the factors.	$The LCD is (x + 1) (x - 3) (x + 3) .$

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.

$The above image shows how to find the LCD (least common denominator) when adding numerical fractions in the example seven-twelfths plus five-eighteenths. The image shows 7 times 3 divided by 12 times 3 plus 5 times 2 plus 18 times 2. Below this is 21 divided by 36 plus 10 divided by 36. The image next to this shows that 12 equals 2 times 2 times 3. Below this shows 18 equals 2 times 3 times 3. A line is drawn. Below it is LCD equals 2 times 2 times 3 times 3. The line below this shows that the LCD equals 36.$

We will do the same thing for rational expressions.

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

Solution

	$Two algebraic fractions are shown: 8 over (x^2 - 2x - 3) and 3x over (x^2 + 4x + 3).$
Factor each denominator.	$Two algebraic fractions are shown: 8 over (x+1)(x-3) and 3x over (x+1)(x+3).$
Find the LCD.
Multiply each denominator by the 'missing' factor and multiply each numerator by the same factor.
Simplify the numerators.	$Two algebraic fractions displayed. The first is 8x+24 over (x+1)(x-3)(x+3), and the second is 3x^2-9x over (x+1)(x+3)(x-3). Both fractions have common denominators.$

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.

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

Solution

	$The image shows the fraction addition problem '7 over 12 plus 5 over 18'.$
Find the LCD of 12 and 18.
Rewrite each fraction as an equivalent fraction with the LCD.	$A math problem displaying the sum of two fractions: (7 * 3) / (12 * 3) + (5 * 2) / (18 * 2). The numbers 3 and 2 are highlighted in red, suggesting multiplication to achieve common denominators.$
Add the fractions.	$A mathematical expression showing the sum of two fractions with a common denominator: 21/36 + 10/36.$
The fraction cannot be simplified.	$A fraction is displayed with 31 as the numerator, a horizontal line, and 36 as the denominator.$

Now we will add rational expressions whose denominators are monomials.

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

Solution

		$An algebraic expression showing the sum of two fractions: 5 over 12x squared y, plus 4 over 21xy squared.$
Find the LCD of $12 x^{2} y$ and $21 x y^{2}$ .
		$A mathematical expression showing the sum of two algebraic fractions: 5/(12x^2y) + 4/(21xy^2).$
Rewrite each rational expression as an equivalent fraction with the LCD.		$An algebraic expression displaying the addition of two rational terms: (5 * 7y) / (12x^2y * 7y) and (4 * 4x) / (21xy^2 * 4x), illustrating a step in finding a common denominator for two fractions.$
Simplify.		$A mathematical expression showing the sum of two fractions: 35y over 84x^2y^2 plus 16x over 84x^2y^2. Both fractions share a common denominator.$
Add the rational expressions.		$A fraction with the numerator 16x + 35y and the denominator 84x^2y^2 is displayed on a white background, representing a mathematical algebraic expression.$
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

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

Solution

$The above image shows the steps to add fractions whose denominators are monomials for the example 5 divided by 12 x squared y plus 4 divided by 21 x y squared. Find the LCD of 12 x squared y and 21 x y squared. To the right of this expression is 12 x squared y equals 2 times 2 times 3 times x times x times y. Below that is 21 x y squared equals 3 times 7 times x times y times y. A line is drawn. Below that is LCD equals 2 times 2 times 3 times 7 times x times x times y times y. Below that is LCD equals 84 x squared y squared. Rewrite each rational expression as an equivalent fraction with the LCD. The original equation is shown. Below that is 5 times 7 y divided by 12 x squared y times 7 y plus 4 times 4 x divided by 21 x y squared times 4 x. Simplify to get 35 y divided by 84 x squared y squared plus 16 x divided by x squared y squared. Add the rational expressions 16 x plus 35 y divided by 84 x squared y squared. There are no factors common to the numeration and denominator. The fraction cannot be simplified.$ Step 2 is to add the rational expression. Then, add the numerators and place the sum over the common denominator to get 3 x minus 6 plus 2 x minus 6 divided by x minus 3 times x minus 2.

Step 2 is to add the rational expression. Then, add the numerators and place the sum over the common denominator to get 3 x minus 6 plus 2 x minus 6 divided by x minus 3 times x minus 2.

Step 3 is to simplify, if possible. Because 5 x minus 12 cannot be factored, the answer is simplified to 5 x minus 12 divided by x minus 3 times x minus 2.

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

Add: $\frac{2 a}{2 a b + b^{2}} + \frac{3 a}{4 a^{2} - b^{2}} .$

Solution

	$A mathematical expression showing the sum of two fractions: 2a divided by (2ab + b^2) plus 3a divided by (4a^2 - b^2).$
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.	$An algebraic expression displaying the sum of two fractions with a common denominator, prior to simplification. Numerators are 2a(2a-b) and 3ab, while the common denominator is b(2a+b)(2a-b).$
Simplify the numerators.	$An algebraic expression showing the sum of two fractions with a common denominator of b(2a+b)(2a-b). The first numerator is 4a^2-2ab, and the second numerator is 3ab.$
Add the rational expressions.	$A mathematical expression showing a fraction with 4a^2 - 2ab + 3ab in the numerator and b(2a + b)(2a - b) in the denominator.$
Simplify the numerator.	$A mathematical fraction is shown, with the numerator as 4a^2 + ab and the denominator as b(2a + b)(2a - b).$
Factor the numerator.	$A mathematical fraction with the numerator a(4a + b) and the denominator b(2a + b)(2a - b).$
There are no factors common to the numerator and denominator. The fraction cannot be simplified.

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

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

Solution

	$An algebraic expression featuring the sum of two rational fractions: 8/(x^2 - 2x - 3) + 3x/(x^2 + 4x + 3).$
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.	$An algebraic expression showing the sum of two rational fractions, where common factors in the numerators and denominators are highlighted in red, indicating potential simplification steps.$
Simplify the numerators.
Add the rational expressions.	$A mathematical fraction displaying the expression (3x^2 - x + 24) divided by (x + 1)(x - 3)(x + 3).$
Simplify the numerator.	$A mathematical fraction displaying the expression (3x^2 - x + 24) divided by (x + 1)(x - 3)(x + 3).$
The numerator is prime, so there are no common factors.

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

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

Solution

The above image has 3 columns. It shows the steps on how to subtract rational expressions with different denominators for x divided by x minus three minus x plus x minus 3. Step 1 is to 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. In the above expression, the answer is no. Find the LCD of x minus 3, x plus 3. To the right of this is x – 3: x – 3. Below that is x – 2: x – 2. A line is drawn. Below that is written the LCD is x – 3 times x plus 3. Rewrite as x times x plus 3 divided by x minus 3 times x plus 3 minus x minus 2 times x minus 3 divided by x plus 3 times x minus 3. Keep the denominators factored! Factor to get x squared plus 3 x divided by x minus 3 times x plus 3 minus x squared minus 5 x plus 6 divided by x minus 3 times x plus 3.

Step 2 is to subtract the rational expressions. Subtract the numerators and place the difference over the common denominator to get x 2 plus 3 x minus x squared minus 5 x plus 6 divided by x minus 3 times x plus 3. Then to x squared plus 3 x minus x squared plus 5 x minus 6 divided by x minus 3 times x plus 3. Be careful with the signs! Then to 8 x minus 6 divided by x minus 3 times x plus 3.

Step 3 is to simplify, if possible. The numerator and denominator have no factors in common. The answer is simplified to 2 times 4 x minus 3 divided by x minus 3 times x plus 3.

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

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

Solution

	$A mathematical expression showing the subtraction of two algebraic fractions: 8y divided by the quantity y squared minus 16, minus 4 divided by the quantity y minus 4.$
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.	$A mathematical expression showing the subtraction of two fractions with a common denominator of (y-4)(y+4). The numerators are 8y and 4(y+4), with parts of the second fraction highlighted in red.$
Simplify the numerators.	$An algebraic expression showing the subtraction of two fractions with a common denominator of (y-4)(y+4). The numerators are 8y and 4y+16.$
Subtract the rational expressions.	$An algebraic expression shown as a fraction with 8y - 4y - 16 in the numerator and (y - 4)(y + 4) in the denominator.$
Simplify the numerators.	$A mathematical expression showing the fraction (4y - 16) / ((y - 4)(y + 4)).$
Factor the numerator to look for common factors.	$A mathematical expression showing the fraction 4(y-4) over (y-4)(y+4).$
Remove common factors.	$A mathematical expression showing the fraction 4(y-4) over (y-4)(y+4), with the common factor (y-4) in both the numerator and denominator crossed out, indicating cancellation.$
Simplify.	$A mathematical expression shows the fraction 4 over (y + 4).$

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

Subtract: $\frac{−3 n - 9}{n^{2} + n - 6} - \frac{n + 3}{2 - n} .$

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.	$An algebraic expression showing the sum of two fractions with a common denominator of (n-2)(n+3). The numerators are (-3n - 9) and (n+3)(n+3), with an (n+3) term highlighted in red.$
Simplify the numerators.	$A mathematical expression showing the addition of two rational algebraic expressions. Both fractions have a common denominator of (n-2)(n+3). The numerators are -3n-9 and n^2+6n+9 respectively.$
Simplify the rational expressions.	$A mathematical fraction. The numerator is -3n - 9 + n^2 + 6n + 9, and the denominator is the product of (n-2) and (n+3).$
Simplify the numerator.	$A mathematical expression showing a fraction with n squared plus 3n in the numerator and the product of (n minus 2) and (n plus 3) in the denominator.$
Factor the numerator to look for common factors.	$A mathematical expression showing the fraction n(p+3) / ((n-2)(p+3)), where the (p+3) terms in both the numerator and denominator are crossed out, indicating they are being canceled.$
Simplify.	$A mathematical expression displaying the fraction n over (n-2).$

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

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

Solution

	$A mathematical expression displays a fraction '5c + 4' over 'c - 2', followed by a subtraction of 3. The expression is (5c + 4)/(c - 2) - 3.$
Write $3$ as $\frac{3}{1}$ to have 2 rational expressions.	$A mathematical expression showing the subtraction of two fractions: (5c + 4) / (c - 2) - 3/1.$
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.	$A mathematical expression displaying the subtraction of two algebraic fractions. The first term is (5c+4)/(c-2), and the second term is 3(c-2) over 1(c-2), with factors in red.$
Simplify.	$A mathematical expression showing the subtraction of two fractions with a common denominator (c-2): (5c+4)/(c-2) - (3c-6)/(c-2).$
Subtract the rational expressions.	$A mathematical expression showing the fraction (5c + 4 - (3c - 6)) divided by (c - 2).$
Simplify.	$A fraction with a numerator of 2c + 10 and a denominator of c - 2 is shown on a white background.$
Factor to check for common factors.	$A fraction with 2(c + 5) in the numerator and c - 2 in the denominator.$
There are no common factors; the rational expression is simplified.

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.

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

Solution


Do the rational expressions have a common denominator? No.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.	$A multi-term mathematical expression involving fractions with the variable 'u'. It shows three rational terms being added and subtracted, all sharing a common denominator of u(u-1) or (u-1)u.$

Write as one rational expression.	$A mathematical expression displaying a fraction. The numerator is 2u^2 + u - 1 - 2u + 1, and the denominator is u(u - 1).$
Simplify.	$A mathematical expression displaying the fraction 2u squared minus u over u multiplied by the quantity u minus 1, all in black font against a white background.$
Factor the numerator, and remove common factors.	$A mathematical fraction with mu(2u-1) in the numerator and mu(u-1) in the denominator.$
Simplify.	$A mathematical expression displaying the fraction (2u-1) divided by (u-1).$

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.

Practice Makes Perfect

In the following exercises, find the LCD.

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

Solution

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

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

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

Solution

$(z - 2) (z + 4) (z + 2)$

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

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

Solution

$(b + 3) (b + 3) (b - 5)$

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

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

Solution

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

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

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

In the following exercises, add.

$\frac{5}{24} + \frac{11}{36}$

Solution

$\frac{37}{72}$

$\frac{7}{30} + \frac{13}{45}$

$\frac{9}{20} + \frac{11}{30}$

Solution

$\frac{49}{60}$

$\frac{8}{27} + \frac{7}{18}$

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

Solution

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

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

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

Solution

$\frac{4 m n + 7}{8 m^{2} n}$

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

$\frac{3}{r + 4} + \frac{2}{r - 5}$

Solution

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

$\frac{4}{s - 7} + \frac{5}{s + 3}$

$\frac{8}{t + 5} + \frac{6}{t - 5}$

Solution

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

$\frac{7}{v + 5} + \frac{9}{v - 5}$

$\frac{5}{3 w - 2} + \frac{2}{w + 1}$

Solution

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

$\frac{4}{2 x + 5} + \frac{2}{x - 1}$

$\frac{2 y}{y + 3} + \frac{3}{y - 1}$

Solution

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

$\frac{3 z}{z - 2} + \frac{1}{z + 5}$

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

Solution

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

$\frac{4}{c d + 3 c} + \frac{1}{d^{2} - 9}$

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

In the following exercises, subtract.

$\frac{t}{t - 6} - \frac{t - 2}{t + 6}$

Solution

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

$\frac{v}{v - 3} - \frac{v - 6}{v + 1}$

$\frac{w + 2}{w + 4} - \frac{w}{w - 2}$

Solution

$\frac{−4 (1 + w)}{(w + 4) (w - 2)}$

$\frac{x - 3}{x + 6} - \frac{x}{x + 3}$

$\frac{y - 4}{y + 1} - \frac{1}{y + 7}$

Solution

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

$\frac{z + 8}{z - 3} - \frac{z}{z - 2}$

$\frac{5 a}{a + 3} - \frac{a + 2}{a + 6}$

Solution

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

$\frac{3 b}{b - 2} - \frac{b - 6}{b - 8}$

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

Solution

$\frac{3}{c - 5}$

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

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

Solution

$\frac{−6}{m - 6}$

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

$\frac{−9 p - 17}{p^{2} - 4 p - 21} - \frac{p + 1}{7 - p}$

Solution

$\frac{p + 2}{p + 3}$

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

$\frac{−2 r - 16}{r^{2} + 6 r - 16} - \frac{5}{2 - r}$

Solution

$\frac{3}{r - 2}$

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

$\frac{5 v - 2}{v + 3} - 4$

Solution

$\frac{v - 14}{v + 3}$

$\frac{6 w + 5}{w - 1} + 2$

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

Solution

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

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

In the following exercises, add and subtract.

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

Solution

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

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

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

Solution

$\frac{c - 5}{c + 2}$

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

In the following exercises, simplify.

$\frac{6 a}{3 a b + b^{2}} + \frac{3 a}{9 a^{2} - b^{2}}$

Solution

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

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

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

Solution

$\frac{3}{d + 8}$

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

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

Solution

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

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

$\frac{−5 n - 5}{n^{2} + n - 6} + \frac{n + 1}{2 - n}$

Solution

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

$\frac{−4 b - 24}{b^{2} + b - 30} + \frac{b + 7}{5 - b}$

$\frac{7}{15 p} + \frac{5}{18 p q}$

Solution

$\frac{42 q + 25}{90 p q}$

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

$\frac{4}{x - 2} + \frac{3}{x + 5}$

Solution

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

$\frac{6}{m + 4} + \frac{9}{m - 8}$

$\frac{2 q + 7}{q + 4} - 2$

Solution

$\frac{- 1}{q + 4}$

$\frac{3 y - 1}{y + 4} - 2$

$\frac{z + 2}{z - 5} - \frac{z}{z + 1}$

Solution

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

$\frac{t}{t - 5} - \frac{t - 1}{t + 5}$

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

Solution

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

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

Everyday Math

Decorating cupcakes Victoria can decorate an order of cupcakes for a wedding in $t$ hours, so in 1 hour she can decorate $\frac{1}{t}$ of the cupcakes. It would take her sister 3 hours longer to decorate the same order of cupcakes, so in 1 hour she can decorate $\frac{1}{t + 3}$ of the cupcakes.

ⓐ Find the fraction of the decorating job that Victoria and her sister, working together, would complete in one hour by adding the rational expressions $\frac{1}{t} + \frac{1}{t + 3} .$
ⓑ Evaluate your answer to part (a) when $t = 5 .$

Solution

ⓐ $\frac{2 t + 3}{t (t + 3)}$ ⓑ $\frac{13}{40}$

Kayaking When Trina kayaks upriver, it takes her $\frac{5}{3 - c}$ hours to go 5 miles, where $c$ is the speed of the river current. It takes her $\frac{5}{3 + c}$ hours to kayak 5 miles down the river.

ⓐ Find an expression for the number of hours it would take Trina to kayak 5 miles up the river and then return by adding $\frac{5}{3 - c} + \frac{5}{3 + c} .$
ⓑ Evaluate your answer to part (a) when $c = 1$ to find the number of hours it would take Trina if the speed of the river current is 1 mile per hour.

Writing Exercises

Felipe thinks $\frac{1}{x} + \frac{1}{y}$ is $\frac{2}{x + y} .$

ⓐ Choose numerical values for x and y and evaluate $\frac{1}{x} + \frac{1}{y} .$
ⓑ Evaluate $\frac{2}{x + y}$ for the same values of x and y you used in part (a).
ⓓ Find the correct expression for $\frac{1}{x} + \frac{1}{y} .$

Solution

Answers may vary.

Simplify the expression $\frac{4}{n^{2} + 6 n + 9} - \frac{1}{n^{2} - 9}$ and explain all your steps.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “find the least common denominator of rational expressions,” “find equivalent rational expressions,” “add rational expressions with different denominators,” and “subtract rational expressions with different denominators.” The rest of the cells are blank.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?