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Section 7.5 Mixed Operations
We can also simplify expressions that have some combinations of addition and subtraction, along with multiplication or division. When doing this, just remember order of operations (BEDMAS). In particular,
Brackets should be done first.
Multiplication and division are next, from left to right.
Addition and subtraction are last, from left to right.
Subsection 7.5.1 Mixed Operations Practice
Exercise Group 7.5.1 . Mixed Operations with Rational Expressions.
Simplify each expression.
(a) \(\dfrac{x^2-2x+1}{x^2+3x+2} \div \dfrac{x^2-1}{x^2-4} + \dfrac{x-2}{x+1}\) Answer .
\(\dfrac{2x(x-2)}{(x+1)^2}\)
(b) \(\dfrac{x^2-1}{x-2} \cdot \dfrac{2}{x+1} - \dfrac{1}{x+2}\) Answer .
\(\dfrac{2x^2+x-2}{(x-2)(x+2)}\)
(c) \(\dfrac{9x-12}{x^2-16} - \dfrac{x+3}{5x^2+15x} \cdot \dfrac{15x^2}{x^2-4x}\)
(d) \(\left(\dfrac{x+3}{x+2}+\dfrac{6}{x-8}\right)\div\dfrac{2x^2+9x+4}{2x^2+5x+2}\)
(e) \(\dfrac{x+1}{2x-6}\div\dfrac{2(x+1)^2}{2-x}+\dfrac{11}{x-2}\) Answer .
\(\dfrac{43x^2-84x-136}{4(x-3)(x+1)(x-2)}\)
(f) \(\dfrac{x+1}{x^2+2x-35} + \dfrac{x^2+x-12}{x^2-2x-24} \cdot \dfrac{x^2-4x-12}{x^2+2x-15}\) Answer .
\(\dfrac{x^3+5x^2-25x-65}{(x+7)(x-5)(x+5)}\)
(g) \(\left(\dfrac{x-7}{x^2-16} - \dfrac{x-1}{16-x^2}\right) \left(\dfrac{x^2-16}{2}\right)\)
(h) \(\dfrac{x+5}{x+6}+\dfrac{1}{x+4}\div\dfrac{x+6}{x^2-x-20}\)
(i) \(\left(\dfrac{x-3}{x^2-9}+\dfrac{x+3}{x^2+6x+9}\right)\dfrac{x+3}{x+1}\)
(j) \(\left(\dfrac{x+3}{x-5}+\dfrac{x-2}{x+4}\right)(x^2-x-20)\)
(k) \(\dfrac{5}{x}-\dfrac{3}{x^3}\div\dfrac{2}{x}\)
(l) \(\dfrac{4}{2x^3}-\dfrac{5x+10}{x^8}\div\dfrac{x+2}{x^3}\)
(m) \(\left(\dfrac{x}{x^2-16}-\dfrac{2}{3x+12}\right)\left(\dfrac{x-4}{6}\right)\)
(n) \(\dfrac{x^3}{3}-\dfrac{2x^2+xy}{xy}\cdot\dfrac{y}{10x+5y}\)
(o) \(\left(\dfrac{x+6}{x+2}-\dfrac{4}{x}\right)\div\dfrac{x^2-16}{x^2-4}\) Answer .
\(\dfrac{(x-2)^2}{x(x-4)}\)
(p) \(\dfrac{2x^2}{x-1}-\dfrac{2x^2-7x+3}{x-3}\cdot\dfrac{x+2}{x-1}\)
Exercise Group 7.5.2 . Mixed Operations with Multiple Variables.
Simplify each expression.
(a) \(\dfrac{2}{x} + \dfrac{x^2-y^2}{4x+4y}\cdot\dfrac{12x^2}{3y-3x}\)
(b) \(\dfrac{2x^3}{3y^2}\cdot\dfrac{9y}{10x}-\dfrac{2y}{3x}\) Answer .
\(\dfrac{9x^3-10y^2}{15xy}\)
(c) \(\dfrac{5m-n}{2m+n}-\dfrac{4m^2-4mn+n^2}{4m^2-n^2}\div\dfrac{6m^2-mn-n^2}{3m+15n}\) Answer .
\(\dfrac{15m^2+2mn-n^2-3m-15n}{(2m+n)(3m+n)}\)
(d) \(\dfrac{2}{b^2}+\dfrac{6ab}{4ab+4b^2}\div\dfrac{7a-7b}{a^2-b^2}\) Answer .
\(\dfrac{3ab^2+28}{14b^2}\)
