For example, to add \(\frac{3}{5} + \frac{1}{5}\text{,}\) we simply add the numerators, because the denominators are the same,
\begin{gather*}
\frac{3}{5} + \frac{1}{5} = \frac{4}{5}
\end{gather*}
Section 7.4 Adding and Subtracting Rational Expressions
Adding rational expressions is similar to adding (numerical) fractions. Recall that to add fractions, you need to have the denominators be equal, and then you add the numerators.
Example 7.4.1. Adding Two Fractions.
In general, if the denominators are the same (that is, we have a common denominator), then simply add the numerators together over that common denominator. That is,
\begin{gather*}
\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}
\end{gather*}
If the denominators are not equal, then we need to first manipulate the fractions so that they have a common denominator.
Example 7.4.2. Adding Two Fractions with Different Denominators.
For example, to add \(\frac{4}{9} + \frac{1}{3}\text{,}\) we have denominators of 9 and 3, so we can make a common denominator of 9, by multiplying \(\frac{1}{3}\) by \(\frac{3}{3}\text{,}\)
\begin{align*}
\frac{4}{9} + \frac{1}{3} \amp = \frac{4}{9} + \frac{1}{3} \cdot \frac{3}{3} \amp\amp \text{making a common denominator}\\
\amp = \frac{4}{9} + \frac{3}{9}\\
\amp = \frac{7}{9} \amp\amp \text{adding numerators}
\end{align*}
Sometimes, to make a common denominator, you need to manipulate both fractions.
Example 7.4.3. Adding Two Fractions with Different Denominators.
For example, to add \(\frac{3}{5} + \frac{2}{7}\text{,}\) we have denominators of 5 and 7, so we need a common denominator of \(35\) (which is \(5 \times 7\)),
\begin{align*}
\frac{3}{5} + \frac{2}{7} \amp = \frac{3}{5} \cdot \frac{7}{7} + \frac{2}{7} \cdot \frac{5}{5} \amp\amp \text{making a common denominator}\\
\amp = \frac{21}{35} + \frac{10}{35}\\
\amp = \frac{31}{35} \amp\amp \text{adding numerators}
\end{align*}
Subsection 7.4.1 Adding Rational Expressions
Example 7.4.6. Adding Rational Expressions with Like Denominators.
To add \(\frac{4}{x+2} + \frac{3}{x+2}\text{,}\) add the numerators over the common denominator of \(x+2\text{,}\)
\begin{gather*}
\frac{4}{x + 2} + \frac{3}{x + 2} = \frac{4 + 3}{x + 2} = \frac{7}{x + 2}
\end{gather*}
Example 7.4.7. Adding Rational Expressions with Like Denominators.
To add \(\frac{2x - 1}{x + 3} + \frac{1 - x}{x + 3}\text{,}\) add the numerators together over the common denominator of \(x+3\text{,}\)
\begin{align*}
\frac{2x - 1}{x + 3} + \frac{1 - x}{x + 3} \amp = \frac{2x - 1 + 1 - x}{x + 3}\\
\amp = \frac{x}{x + 3}
\end{align*}
To add two rational expressions with like denominators, add their numerators and keep the same denominator. That is, if \(A, B, C\) are polynomials, then,
\begin{gather*}
\boxed{\frac{A}{C} + \frac{B}{C} = \frac{A + B}{C}}
\end{gather*}
with the restriction that \(C\) cannot be equal to 0.
Sometimes, after adding, the result can be further simplified by cancelling common factors.
Example 7.4.8. Adding Rational Expressions and Simplifying.
For example,
\begin{align*}
\frac{4x}{15} + \frac{8x}{15} \amp = \frac{4x + 8x}{15} \amp\amp \text{adding}\\
\amp = \frac{12x}{15} \amp\amp \text{collecting like terms}\\
\amp = \frac{4x}{5} \amp\amp \text{cancelling common factor of 3}
\end{align*}
Subsection 7.4.2 Summary of Adding or Subtracting Rational Expressions
-
Factor the denominators completely.
-
Create a common denominator, by manipulating each fraction.
-
After you have a common denominator, add or subtract the numerators, and keep that common denominator.
-
Simplify if needed, by collecting like terms, and factor and cancel if possible.
Note: \(\frac{A}{B+C} \neq \frac{A}{B} + \frac{A}{C}\text{.}\) For example, \(\frac{2}{3} \neq \frac{2}{2} + \frac{2}{1} = 3\text{.}\)
Subsection 7.4.3 Adding and Subtracting with Monomial Denominators
Exercise Group 7.4.1. Monomial Like Denominators.
Find each sum or difference, and simplify.
(a)
\(\dfrac{x}{8}+\dfrac{3x}{8}\)Answer.
\(\dfrac{x}{2}\)
(b)
\(\dfrac{1}{3x}+\dfrac{5}{3x}\)Answer.
\(\dfrac{2}{x}\)
(c)
\(\dfrac{7x}{16}-\dfrac{5x}{16}\)Answer.
\(\dfrac{x}{8}\)
(d)
\(\dfrac{7x^2}{4x^3}-\dfrac{x^2}{4x^3}\)Answer.
\(\dfrac{3}{2x}\)
(e)
\(\dfrac{-8x^2}{3x}+\dfrac{11x^2}{3x}\)Answer.
\(x\)
(f)
\(\dfrac{x+9}{10}+\dfrac{x+7}{10}\)Answer.
\(\dfrac{x+8}{5}\)
(g)
\(\dfrac{5x}{15}+\dfrac{7x}{15}\)Answer.
\(\dfrac{4x}{5}\)
(h)
\(\dfrac{11x}{12x^2}-\dfrac{7x}{12x^2}\)Answer.
\(\dfrac{1}{3x}\)
(i)
\(\dfrac{5x+2}{x^2}-\dfrac{4x+2}{x^2}\)Answer.
\(\dfrac{1}{x}\)
(j)
\(\dfrac{7x+1}{x}+\dfrac{5x-2}{x}\)Answer.
\(\dfrac{12x-1}{x}\)
(k)
\(\dfrac{11x}{6}-\dfrac{4x}{6}\)Answer.
\(\dfrac{7x}{6}\)
(l)
\(\dfrac{7}{x}+\dfrac{3}{x}\)Answer.
\(\dfrac{10}{x}\)
(m)
\(\dfrac{5x+3}{10}+\dfrac{3x+5}{10}\)Answer.
\(\dfrac{4(x+1)}{5}\)
(n)
\(\dfrac{x}{6}+\dfrac{2x}{6}\)Answer.
\(\dfrac{x}{2}\)
Exercise Group 7.4.2. Monomial Different Denominators.
Find each sum or difference, and simplify.
(a)
\(\dfrac{x}{3}+\dfrac{x}{2}\)Answer.
\(\dfrac{5x}{6}\)
(b)
\(\dfrac{x+10}{6}-\dfrac{2x-1}{30}\)Answer.
\(\dfrac{x+17}{10}\)
(c)
\(\dfrac{1}{3x}+\dfrac{2}{5x}\)Answer.
\(\dfrac{11}{15x}\)
(d)
\(\dfrac{4}{5x}-\dfrac{7}{15x}\)Answer.
\(\dfrac{1}{3x}\)
(e)
\(\dfrac{9}{4x}+\dfrac{3}{2x}\)Answer.
\(\dfrac{15}{4x}\)
(f)
\(\dfrac{5x}{3}-\dfrac{x}{2}\)Answer.
\(\dfrac{7x}{6}\)
(g)
\(\dfrac{2x}{5}+\dfrac{6x}{2}\)Answer.
\(\dfrac{17x}{5}\)
(h)
\(\dfrac{x-2}{3}+\dfrac{x+1}{6}\)Answer.
\(\dfrac{x-1}{2}\)
(i)
\(\dfrac{3}{4x}+\dfrac{1}{20x}\)Answer.
\(\dfrac{4}{5x}\)
(j)
\(\dfrac{x-1}{2}-\dfrac{3x+1}{18}\)Answer.
\(\dfrac{3x-5}{9}\)
(k)
\(\dfrac{3}{2x}+\dfrac{1}{6}\)Answer.
\(\dfrac{x+9}{6x}\)
(l)
\(4-\dfrac{6}{5x}\)Answer.
\(\dfrac{20x-6}{5x}\)
(m)
\(\dfrac{3x+2}{2}+\dfrac{4x+5}{5}\)Answer.
\(\dfrac{23x+20}{10}\)
Exercise Group 7.4.3. Single Variable.
Find each sum or difference, and simplify.
(a)
\(\dfrac{x}{2}-\dfrac{x}{5}-\dfrac{x}{6}\)Answer.
\(\dfrac{2x}{15}\)
(b)
\(\dfrac{9x}{5}-\dfrac{3x}{4}+\dfrac{7x}{10}\)Answer.
\(\dfrac{7x}{4}\)
(c)
\(\dfrac{5x+9}{12}-\dfrac{2x+8}{16}\)Answer.
\(\dfrac{7x+6}{24}\)
(d)
\(\dfrac{7x}{6}-\dfrac{x}{2}+\dfrac{2x}{9}\)Answer.
\(\dfrac{8x}{9}\)
(e)
\(\dfrac{5}{6x^2}+\dfrac{4}{3x}\)Answer.
\(\dfrac{8x+5}{6x^2}\)
(f)
\(\dfrac{3}{8x^2}+\dfrac{1}{4x}-\dfrac{5}{6x^3}\)Answer.
\(\dfrac{6x^2+9x-20}{24x^3}\)
(g)
\(\dfrac{5}{4x^2}+\dfrac{1}{7x^3}\)Answer.
\(\dfrac{35x+4}{28x^3}\)
(h)
\(\dfrac{2}{x}+\dfrac{6}{x^2}\)Answer.
\(\dfrac{2(x+3)}{x^2}\)
(i)
\(\dfrac{5}{3x^2}-\dfrac{7}{5}\)Answer.
\(\dfrac{25-21x^2}{15x^2}\)
(j)
\(\dfrac{7}{6x^2}-\dfrac{3}{8x^3}\)Answer.
\(\dfrac{28x-9}{24x^3}\)
(k)
\(\dfrac{2x}{3}+\dfrac{3x}{4}-\dfrac{x}{6}\)Answer.
\(\dfrac{5x}{4}\)
(l)
\(\dfrac{5}{x^2}-\dfrac{3}{4x^3}\)Answer.
\(\dfrac{20x-3}{4x^3}\)
(m)
\(\dfrac{3}{x^4}+\dfrac{1}{2x^2}-\dfrac{3}{5x}\)Answer.
\(\dfrac{-6x^3+5x^2+30}{10x^4}\)
(n)
\(\dfrac{17}{6x}-\dfrac{11}{2x}+\dfrac{1}{3x}\)Answer.
\(-\dfrac{7}{3x}\)
Exercise Group 7.4.4. Multiple Variables.
Find each sum or difference, and simplify.
(a)
\(\dfrac{2a}{b}-\dfrac{a-1}{b}\)Answer.
\(\dfrac{a+1}{b}\)
(b)
\(\dfrac{4z}{xy}-\dfrac{9x}{yz}\)Answer.
\(\dfrac{4z^2-9x^2}{xyz}\)
(c)
\(\dfrac{7ab}{10a^2b}-\dfrac{3ab}{10a^2b}\)Answer.
\(\dfrac{2}{5a}\)
(d)
\(\dfrac{m}{n}-\dfrac{m+1}{n}\)Answer.
\(-\dfrac{1}{n}\)
(e)
\(\dfrac{10x^3y}{7xy^2}+\dfrac{4x^3y}{7xy^2}\)Answer.
\(\dfrac{2x^2}{y}\)
(f)
\(\dfrac{2x}{xy}+\dfrac{4}{x^2}-3\)Answer.
\(\dfrac{2x^2+4y-3x^2y}{x^2y}\)
(g)
\(\dfrac{5}{3x^2y^3}-\dfrac{1}{6xy^4}\)Answer.
\(\dfrac{-x+10y}{6x^2y^4}\)
(h)
\(\dfrac{2s}{5t^2}+\dfrac{1}{10t}-\dfrac{6}{15t^3}\)Answer.
\(\dfrac{4st+t^2-4}{10t^3}\)
(i)
\(\dfrac{6xy}{a^2b}-\dfrac{2x}{ab^2y}+1\)Answer.
\(\dfrac{a^2b^2y-2ax+6bxy^2}{a^2b^2y}\)
(j)
\(\dfrac{6}{3xy}-\dfrac{5}{y^2}\)Answer.
\(-\dfrac{5x-2y}{xy^2}\)
(k)
\(\dfrac{2x}{3y}-\dfrac{x^2}{4y^3}+\dfrac{3}{5y^4}\)Answer.
\(\dfrac{-15x^2y+40xy^3+36}{60y^4}\)
(l)
\(\dfrac{n}{m}+\dfrac{m}{n}-m\)Answer.
\(\dfrac{m^2+n^2-m^2n}{mn}\)
(m)
\(\dfrac{2x^3}{3y^2}\cdot\dfrac{9y}{10x}-\dfrac{2y}{3x}\)Answer.
\(\dfrac{9x^3-10y^2}{15xy}\)
Subsection 7.4.4 Adding and Subtracting with Like Denominators
You should think of each binomial as a single factor.
Exercise Group 7.4.5. Binomial Like Denominators.
Find each sum or difference, and simplify.
(a)
\(\dfrac{5x}{x+4}+\dfrac{20}{x+4}\)Answer.
\(5\)
(b)
\(\dfrac{9x}{3x-4}-\dfrac{12}{3x-4}\)Answer.
\(3\)
(c)
\(\dfrac{x+2}{2x-1}+\dfrac{3x-3}{2x-1}\)Answer.
\(\dfrac{4x-1}{2x-1}\)
(d)
\(\dfrac{x-4}{x+1}+\dfrac{x+6}{x+1}\)Answer.
\(2\)
(e)
\(\dfrac{3x}{x+1}+\dfrac{3}{x+1}\)Answer.
\(3\)
(f)
\(\dfrac{6x}{x-2}-\dfrac{12}{x-2}\)Answer.
\(6\)
(g)
\(\dfrac{14x}{2x+3y}+\dfrac{21y}{2x+3y}\)Answer.
\(7\)
(h)
\(\dfrac{x-2}{5x+3}-\dfrac{6x-5}{5x+3}\)Answer.
\(\dfrac{3-5x}{5x+3}\)
(i)
\(\dfrac{9x-1}{3x-5}-\dfrac{3x+9}{3x-5}\)Answer.
\(2\)
(j)
\(\dfrac{-15x}{1-5x}-\dfrac{3}{5x-1}\)Answer.
\(3\)
(k)
\(\dfrac{2x}{3x-1}+\dfrac{x}{1-3x}\)Answer.
\(\dfrac{x}{3x-1}\)
(l)
\(\dfrac{8x}{6x-5}+\dfrac{10-4x}{5-6x}\)Answer.
\(2\)
(m)
\(\dfrac{x^2-8x}{x-5}-\dfrac{15}{5-x}\)Answer.
\(x-3\)
(n)
\(\dfrac{x}{x-3}-\dfrac{3}{x-3}\)Answer.
\(1\)
(o)
\(\dfrac{2x}{x+4}+\dfrac{8}{x+4}\)Answer.
\(2\)
(p)
\(\dfrac{x^2}{x-2}+\dfrac{3x}{x-2}-\dfrac{10}{x-2}\)Answer.
\(x+5\)
(q)
\(\dfrac{10x-1}{4x-3}-\dfrac{8-2x}{4x-3}\)Answer.
\(3\)
(r)
\(\dfrac{x^2}{x+1}+\dfrac{x}{x+1}\)Answer.
\(x\)
(s)
\(\dfrac{x^2}{x-4}-\dfrac{x}{x-4}-\dfrac{12}{x-4}\)Answer.
\(x+3\)
(t)
\(\dfrac{2x}{x - 2} - \dfrac{4}{x - 2}\)Answer.
\(2\)
Exercise Group 7.4.6. Quadratic Denominators.
Find each sum or difference, and simplify.
(a)
\(\dfrac{x+4}{x^2-2x-3}+\dfrac{6x+3}{x^2-2x-3}\)Answer.
\(\dfrac{7}{x-3}\)
(b)
\(\dfrac{6x+3}{x^2+6x+5}-\dfrac{x-2}{x^2+6x+5}\)Answer.
\(\dfrac{5}{x+5}\)
(c)
\(\dfrac{6x+4}{4x^2+4x}+\dfrac{x+3}{4x^2+4x}\)Answer.
\(\dfrac{7}{4x}\)
(d)
\(\dfrac{3x+2}{x^2-25}-\dfrac{2x-3}{x^2-25}\)Answer.
\(\dfrac{1}{x-5}\)
(e)
\(\dfrac{7x + 9}{x^2 + 3x + 4} - \dfrac{2x - 7}{x^2 + 3x + 4}\)Answer.
\(\dfrac{5x + 16}{x^2 + 3x + 4}\)
(f)
\(\dfrac{x^2+5x}{4x^2+5x-6}-\dfrac{3x}{4x^2+5x-6}\)Answer.
\(\dfrac{x}{4x-3}\)
(g)
\(\dfrac{x}{x^2-1}-\dfrac{1}{x^2-1}\)Answer.
\(\dfrac{1}{x+1}\)
(h)
\(\dfrac{3x-2}{x^2-25}-\dfrac{4x-7}{x^2-25}\)Answer.
\(-\dfrac{1}{x+5}\)
(i)
\(\dfrac{7x - 12}{2x^2 + 5x - 12} - \dfrac{3x - 6}{2x^2 + 5x - 12}\)Answer.
\(\dfrac{2}{x + 4}\)
(j)
\(\dfrac{2x-5}{x^2-9}-\dfrac{3x-8}{x^2-9}\)Answer.
\(-\dfrac{1}{x+3}\)
(k)
\(\dfrac{2x^2+1}{2x^2-5x-12}-\dfrac{4-x}{2x^2-5x-12}\)Answer.
\(\dfrac{x-1}{x-4}\)
(l)
\(\dfrac{2x^2-x}{(x-3)(x+1)}+\dfrac{3-6x}{(x-3)(x+1)}-\dfrac{8}{(x-3)(x+1)}\)Answer.
\(\dfrac{2x^2-7x-5}{(x-3)(x+1)}\)
(m)
\(\dfrac{2x^2}{x^2 - x - 12} - \dfrac{7x + 4}{x^2 - x - 12}\)Answer.
\(\dfrac{2x + 1}{x + 3}\)
Subsection 7.4.5 Adding and Subtracting with Polynomials (Factoring)
Exercise Group 7.4.7. Binomial Denominators.
Find each sum or difference, and simplify.
(a)
\(\dfrac{x+24}{5x+20}+\dfrac{x}{x+4}\)Answer.
\(\dfrac{6}{5}\)
(b)
\(\dfrac{1}{x-7}+\dfrac{6}{2x-14}\)Answer.
\(\dfrac{4}{x-7}\)
(c)
\(\dfrac{12}{4x-20}-\dfrac{1}{x-5}\)Answer.
\(\dfrac{2}{x-5}\)
(d)
\(\dfrac{x-4}{3x-3}+\dfrac{1}{x-1}\)Answer.
\(\dfrac{1}{3}\)
(e)
\(\dfrac{3x}{3x+6}+\dfrac{1}{x+2}\)Answer.
\(\dfrac{x+1}{x+2}\)
(f)
\(\dfrac{7}{x-2}+\dfrac{4}{2-x}\)Answer.
\(\dfrac{3}{x-2}\)
(g)
\(\dfrac{4x}{x-1}-4\)Answer.
\(\dfrac{4}{x-1}\)
(h)
\(\dfrac{5}{x-2}-\dfrac{3}{x+2}\)Answer.
\(\dfrac{2x+16}{x^2-4}\)
(i)
\(\dfrac{10}{3x-12}-\dfrac{3}{x-4}\)Answer.
\(\dfrac{1}{3(x-4)}\)
(j)
\(\dfrac{3x}{x+2}-\dfrac{x}{x-2}\)Answer.
\(\dfrac{2x(x-4)}{x^2-4}\)
(k)
\(\dfrac{3x}{2x+1}+\dfrac{4}{x-3}\)Answer.
\(\dfrac{3x^2-x+4}{(x-3)(2x+1)}\)
(l)
\(\dfrac{3}{x-3}-\dfrac{7}{5x-1}\)Answer.
\(\dfrac{2(4x+9)}{(x-3)(5x-1)}\)
(m)
\(\dfrac{7}{x-4}+\dfrac{2}{x}\)Answer.
\(\dfrac{9x-8}{x(x-4)}\)
(n)
\(\dfrac{4}{3x-2}+6\)Answer.
\(\dfrac{2(9x-4)}{3x-2}\)
(o)
\(\dfrac{5}{x+4}+\dfrac{7}{x+3}\)Answer.
\(\dfrac{12x+43}{(x+3)(x+4)}\)
(p)
\(\dfrac{6}{2x-3}-\dfrac{4}{x-5}\)Answer.
\(-\dfrac{2(x+9)}{(x-5)(2x-3)}\)
(q)
\(\dfrac{7x}{x+4}+\dfrac{3x}{x-6}\)Answer.
\(\dfrac{10x(x-3)}{(x-6)(x+4)}\)
(r)
\(\dfrac{7}{2x-6}+\dfrac{4}{10x-15}\)Answer.
\(\dfrac{3(26x-43)}{10(x-3)(2x-3)}\)
(s)
\(\dfrac{2}{x+1}-\dfrac{3}{x-2}\)Answer.
\(-\dfrac{x+7}{(x+1)(x-2)}\)
Exercise Group 7.4.8. Quadratic Denominators and Factoring.
Find each sum or difference, and simplify.
(a)
\(\dfrac{4x}{x-1}-\dfrac{28x}{x^2+5x-6}\)Answer.
\(\dfrac{4x}{x+6}\)
(b)
\(\dfrac{6x}{x^2-9}+\dfrac{x}{x+3}\)Answer.
\(\dfrac{x}{x-3}\)
(c)
\(\dfrac{x}{x+2}-\dfrac{8}{x^2-4}\)Answer.
\(\dfrac{x-4}{x-2}\)
(d)
\(\dfrac{1}{(x-3)(x+1)}-\dfrac{4}{x+1}\)Answer.
\(\dfrac{13-4x}{(x-3)(x+1)}\)
(e)
\(\dfrac{1}{x-5}+\dfrac{2x-19}{x^2-x-20}\)Answer.
\(\dfrac{3}{x+4}\)
(f)
\(\dfrac{x^2+4x}{x^2+6x+8}+\dfrac{3}{x+2}\)Answer.
\(\dfrac{x+3}{x+2}\)
(g)
\(\dfrac{4x}{x^2+6x+5}+\dfrac{1}{x+1}\)Answer.
\(\dfrac{5}{x+5}\)
(h)
\(\dfrac{x}{x^2 + x - 2} - \dfrac{1}{x + 2}\)Answer.
\(\dfrac{1}{(x+2)(x-1)}\)
(i)
\(\dfrac{5}{x^2-x-6}+\dfrac{1}{x+2}\)Answer.
\(\dfrac{1}{x-3}\)
(j)
\(\dfrac{x}{x-8}-\dfrac{6x+80}{x^2-64}\)Answer.
\(\dfrac{x+10}{x+8}\)
(k)
\(\dfrac{6x}{x^2-4}-\dfrac{3}{x-2}\)Answer.
\(\dfrac{3}{x+2}\)
(l)
\(\dfrac{3x}{x^2-4x}-\dfrac{1}{x-4}\)Answer.
\(\dfrac{2}{x-4}\)
(m)
\(\dfrac{x}{x^2 - 3x - 40} - \dfrac{3}{x - 8}\)Answer.
\(\dfrac{-2x - 15}{(x - 8)(x + 5)}\)
(n)
\(\dfrac{3}{x+1}-\dfrac{6}{x^2+4x+3}\)Answer.
\(\dfrac{3}{x+3}\)
(o)
\(\dfrac{1}{x-6}+\dfrac{x-17}{x^2-x-30}\)Answer.
\(\dfrac{2}{x+5}\)
(p)
\(\dfrac{8x-26}{x^2-4x-21}-\dfrac{3}{x-7}\)Answer.
\(\dfrac{5}{x+3}\)
(q)
\(\dfrac{x^2-3x-1}{2x^2+5x+2}+\dfrac{x}{2x+1}\)Answer.
\(\dfrac{x-1}{x+2}\)
(r)
\(\dfrac{4x}{x^2-36}-\dfrac{2}{x-6}\)Answer.
\(\dfrac{2}{x+6}\)
(s)
\(\dfrac{3x}{x^2-49}-\dfrac{3}{2x-14}\)Answer.
\(\dfrac{3}{2(x+7)}\)
(t)
\(\dfrac{x}{x^2-9}+\dfrac{3}{3-x}\)Answer.
\(-\dfrac{2x+9}{x^2-9}\)
(u)
\(\dfrac{x+1}{x^2-x-6}-\dfrac{2}{x-3}\)Answer.
\(-\dfrac{x+3}{(x-3)(x+2)}\)
(v)
\(\dfrac{4x}{x^2-5x}-\dfrac{3}{2x-10}\)Answer.
\(\dfrac{5}{2(x-5)}\)
(w)
\(\dfrac{1}{x-5}-\dfrac{x}{x^2-x-20}\)Answer.
\(\dfrac{4}{(x+4)(x-5)}\)
(x)
\(\dfrac{x^2-20}{x^2-4}+\dfrac{x-2}{x+2}\)Answer.
\(\dfrac{2(x-4)}{x-2}\)
(y)
\(\dfrac{4}{x^2-1}+\dfrac{3}{x+1}\)Answer.
\(\dfrac{3x+1}{(x-1)(x+1)}\)
(z)
\(\dfrac{8}{x^2-4}-\dfrac{5}{x+2}\)Answer.
\(\dfrac{18-5x}{x^2-4}\)
(aa)
\(\dfrac{1}{x^2-x-12}+\dfrac{3}{x+3}\)Answer.
\(\dfrac{3x-11}{(x-4)(x+3)}\)
(ab)
\(\dfrac{2}{x+3}+\dfrac{7}{x^2-9}\)Answer.
\(\dfrac{2x+1}{x^2-9}\)
(ac)
\(\dfrac{3}{x+1}+\dfrac{4}{x^2-3x-4}\)Answer.
\(\dfrac{3x-8}{(x+1)(x-4)}\)
(ad)
\(\dfrac{2x}{x-4}-\dfrac{5x}{x^2-16}\)Answer.
\(\dfrac{x(2x+3)}{(x-4)(x+4)}\)
(ae)
\(\dfrac{x}{2x - 4} + \dfrac{x - 4}{x^2 - 2x}\)Answer.
\(\dfrac{x + 4}{2x}\)
(af)
\(\dfrac{x}{4x + 28} + \dfrac{9x + 3}{x^2 + 7x}\)Answer.
\(\dfrac{x^2 + 36x + 12}{4x(x + 7)}\)
(ag)
\(\dfrac{x}{x+3} - \dfrac{x^2}{x^2+x-6}\)Answer.
\(\dfrac{-2x}{(x+3)(x-2)}\)
Exercise Group 7.4.9. Polynomial Denominators.
Find each sum or difference, and simplify.
(a)
\(\dfrac{1}{x^2-1}-\dfrac{2}{x^2+x}\)Answer.
\(\dfrac{-x+2}{x(x-1)(x+1)}\)
(b)
\(\dfrac{3x+9}{x^2+7x+10}+\dfrac{14}{x^2+3x-10}\)Answer.
\(\dfrac{3x+2}{x^2-4}\)
(c)
\(\dfrac{x+2}{x^2+x-2}+\dfrac{3}{x^2-1}\)Answer.
\(\dfrac{x+4}{x^2-1}\)
(d)
\(\dfrac{3x+3}{x^2+5x+4}-\dfrac{x-3}{x^2+x-12}\)Answer.
\(\dfrac{2}{x+4}\)
(e)
\(\dfrac{3}{x^2-9}+\dfrac{4}{x^2-6x+9}\)Answer.
\(\dfrac{7x+3}{(x-3)^2(x+3)}\)
(f)
\(\dfrac{3}{x^2+x-6}+\dfrac{5}{x^2+6x+9}\)Answer.
\(\dfrac{8x-1}{(x+3)^2(x-2)}\)
(g)
\(\dfrac{x-1}{2x^2+3x+1}-\dfrac{x+1}{2x^2-x-1}\)Answer.
\(-\dfrac{4x}{(x-1)(x+1)(2x+1)}\)
(h)
\(\dfrac{x}{2x^2+x-1}+\dfrac{3}{3x^2+2x-1}\)Answer.
\(\dfrac{3x^2+5x-3}{(x+1)(2x-1)(3x-1)}\)
(i)
\(\dfrac{4}{4x-1}+\dfrac{8x-15}{4x^2+11x-3}\)Answer.
\(\dfrac{3}{x+3}\)
(j)
\(\dfrac{x-1}{x^2+x-6}-\dfrac{x-2}{x^2+4x+3}\)Answer.
\(\dfrac{4x-5}{(x+3)(x-2)(x+1)}\)
(k)
\(\dfrac{x-5}{x^2+8x-20}+\dfrac{2x+1}{x^2-4}\)Answer.
\(\dfrac{3x(x+6)}{(x-2)(x+2)(x+10)}\)
(l)
\(\dfrac{2}{x^2+x-6}+\dfrac{3}{x^3+2x^2-3x}\)Answer.
\(\dfrac{(x+2)(2x-3)}{x(x-2)(x-1)(x+3)}\)
(m)
\(\dfrac{3x+15}{x^2-25}+\dfrac{4x^2-1}{2x^2+9x-5}\)Answer.
\(\dfrac{2(x^2-3x+5)}{x^2-25}\)
(n)
\(\dfrac{2x}{x^3+x^2-6x}-\dfrac{x-8}{x^2-5x-24}\)Answer.
\(\dfrac{4-x}{(x+3)(x-2)}\)
(o)
\(\dfrac{x+3}{x^2-5x+6}+\dfrac{6}{x^2-7x+12}\)Answer.
\(\dfrac{x+8}{(x-4)(x-2)}\)
(p)
\(\dfrac{2x}{x^2+5x+6}-\dfrac{x-6}{x^2+6x+8}\)Answer.
\(\dfrac{x+9}{(x+3)(x+4)}\)
(q)
\(\dfrac{2x}{x^2-1}-\dfrac{x+2}{x^2+3x-4}\)Answer.
\(\dfrac{x^2+5x-2}{(x-1)(x+1)(x+4)}\)
(r)
\(\dfrac{4x}{x^2+6x+8}-\dfrac{3x}{x^2-3x-10}\)Answer.
\(\dfrac{x(x-32)}{(x+2)(x+4)(x-5)}\)
(s)
\(\dfrac{x-1}{x^2-9}+\dfrac{x+7}{x^2-5x+6}\)Answer.
\(\dfrac{2x^2+7x+23}{(x-3)(x+3)(x-2)}\)
(t)
\(\dfrac{2x+1}{2x^2-14x+24}+\dfrac{5x}{4x^2-8x-12}\)Answer.
\(\dfrac{9x^2-14x+2}{4(x-3)(x-4)(x+1)}\)
(u)
\(\dfrac{3}{4x^2+7x+3}-\dfrac{5}{16x^2+24x+9}\)Answer.
\(\dfrac{7x+4}{(x+1)(4x+3)^2}\)
(v)
\(\dfrac{x-1}{x^2-8x+15}-\dfrac{x-2}{2x^2-9x-5}\)Answer.
\(\dfrac{x^2+4x-7}{(x-5)(x-3)(2x+1)}\)
(w)
\(\dfrac{3x+2}{4x^2-1}+\dfrac{2x-5}{4x^2+4x+1}\)Answer.
\(\dfrac{10x^2-5x+7}{(2x-1)(2x+1)^2}\)
(x)
\(\dfrac{2x}{x^2+x-6}+\dfrac{5}{x^2+2x-8}\)Answer.
\(\dfrac{(2x+3)(x+5)}{(x-2)(x+3)(x+4)}\)
Exercise Group 7.4.10. Complicated Denominators.
Find each sum or difference, and simplify.
(a)
\(\dfrac{5}{x+1}-\dfrac{1}{x}-\dfrac{x-4}{x^2+x}\)Answer.
\(\dfrac{3}{x}\)
(b)
\(\dfrac{4}{x^2-4}+\dfrac{1}{2-x}-\dfrac{1}{x+2}\)Answer.
\(-\dfrac{2}{x+2}\)
(c)
\(\dfrac{5x}{x+2}-\dfrac{x}{x-1}+\dfrac{3}{x^2+x-2}\)Answer.
\(\dfrac{4x-3}{x+2}\)
(d)
\(\dfrac{x-13}{x+2}+\dfrac{x^2-1}{x^2-4x-12}-\dfrac{2x-5}{x-6}\)Answer.
\(-\dfrac{3(6x-29)}{(x-6)(x+2)}\)
(e)
\(\dfrac{5}{2x^3}-\dfrac{3x-9}{x^2-6x+9}+\dfrac{12x}{4x^2-12x}\)Answer.
\(\dfrac{5}{2x^3}\)
(f)
\(\dfrac{2x}{x^2-9}+\dfrac{x}{x^2+6x+9}-\dfrac{3}{x-3}\)Answer.
\(-\dfrac{3(5x+9)}{(x-3)(x+3)^2}\)
(g)
\(\dfrac{9x-12}{x^2-16}-\dfrac{x+3}{5x^2+15x}+\dfrac{15x^2}{x^2-4x}\)Answer.
\(\dfrac{75x^3+344x^2-60x+16}{5x(x-4)(x+4)}\)
(h)
\(\dfrac{2}{x+5}+\dfrac{5x}{x^2-25}+\dfrac{4}{5-x}\)Answer.
\(\dfrac{3(x-10)}{x^2-25}\)
(i)
\(\dfrac{2x+11}{x^2+x-6}-\dfrac{2}{x+3}+\dfrac{3}{2-x}\)Answer.
\(-\dfrac{3}{x+3}\)
(j)
\(\dfrac{5}{x^2-4x+3}-\dfrac{9}{x^2-2x+1}\)Answer.
\(-\dfrac{2(2x-11)}{(x-3)(x-1)^2}\)
(k)
\(\dfrac{2x}{4x^2-9} + \dfrac{x}{2x^5 + 5x - 3} - \dfrac{1}{2x-3}\)Answer.
\(\dfrac{x}{2x^5+5x-3}\)
(l)
\(\dfrac{2}{x^2+x-6} + \dfrac{3}{x^3+2x^2-3x}\)Answer.
\(\dfrac{(2x-3)(x+2)}{x(x-1)(x-2)(x+3)}\)
Subsection 7.4.6 Adding and Subtracting with Two Variables
Exercise Group 7.4.11. Two Variables.
Find each sum or difference, and simplify.
(a)
\(\dfrac{x}{x-y}-\dfrac{y}{x-y}\)Answer.
\(1\)
(b)
\(\dfrac{x^2}{x^2-y^2}-\dfrac{y^2}{y^2-x^2}+\dfrac{2xy}{y^2-x^2}\)Answer.
\(\dfrac{x-y}{x+y}\)
