Unit 0 - Required Prior Knowledge
Required Prior Knowledge
Questions
Calculate Simplify the following
a) \(\frac{24}{28} \)
b) \(\frac{16}{22}\)
c) \(\frac{32}{12}\)
Compute these calculations
d) \(\frac{1}{4}+\frac{3}{8}\)
e) \(\frac{4}{7}+\frac{2}{3}\)
f) \(\frac{5}{4}-\frac{7}{15}\)
g) \(\frac{3}{5}\times\frac{7}{6}\)
h) \(\frac{2}{5}\div\frac{5}{3}\)
i) \(\frac{8}{9}\div\frac{5}{18}\)
Solutions
Get Ready
Questions
Describe the steps required to simplify and perform operations with fractions.
Notes
To simplify an algebraic fraction we must look for common factors in the numerator and denominator that can ‘cancel’.
Examples and Your Turns
Example
Simplify$$\frac{4x}{20}$$
Your Turn
Simplify$$\frac{20}{4x}\\ \frac{4x}{20x}\\ \frac{4x^{3}}{20x}\\ \frac{4x^{2}}{20x^{5}}\\ \frac{yx^{2}}{20y}\\ \frac{yx^{3}}{9y}\\ \frac{5x^{3}y^{2}}{15xy}\\ \frac{6r^{2}t}{15rt^{2}}\\ \frac{y}{y-5}\\ \frac{a^{2}b}{a+b}\\ \frac{x^{2}+y}{3xy}$$
Further Practice
Simplify
(a) \(\frac{3a}{6ab}\)
\(= \frac{1}{2b}\)
(b) \(\frac{15ab}{25bc}\)
\(= \frac{3a}{5c}\)
(c) \(\frac{abc^3}{a^3b^2c}\)
\(= \frac{c^2}{a^2b}\)
(d) \(\frac{4m^3n^2p^5}{6m^2np^3}\)
\(= \frac{2mnp^2}{3}\)
(e) \(\frac{4a^2}{16ab}\)
\(= \frac{a}{4b}\)
(f) \(\frac{21x^2y^2}{28y^2z^2}\)
\(= \frac{3x^2}{4z^2}\)
(g) \(\frac{3x^2y^3}{5xy^2z^4}\)
\(= \frac{3xy}{5z^4}\)
(h) \(\frac{15ax^3y^2}{25a^2xy^6}\)
\(= \frac{3x^2}{5ay^4}\)
(i) \(\frac{2xy^2}{5x^2y}\)
\(= \frac{2y}{5x}\)
(j) \(\frac{8a^2b}{12b^2c}\)
\(= \frac{2a^2}{3bc}\)
(k) \(\frac{2xy^3z^4}{4x^2yz^2}\)
\(= \frac{y^2z^2}{2x}\)
(l) \(\frac{39a^2b^4c^3}{52a^3b^5c^4}\)
\(= \frac{3}{4abc}\)
(m) \(\frac{3abc}{15a^2b^2c}\)
\(= \frac{1}{5ab}\)
(n) \(\frac{12mn^2p}{15m^2np^2}\)
\(= \frac{4n}{5mp}\)
(o) \(\frac{5a^3b^2c^4}{15ab^4c}\)
\(= \frac{a^2c^3}{3b^2}\)
(p) \(\frac{38k^2p^3m^4}{57k^3pm^2}\)
\(= \frac{2p^2m^2}{3k}\)
(q) \(\frac{x^2yz^3}{x^3y^2z}\)
\(= \frac{z^2}{xy}\)
(r) \(\frac{15m^2p^3}{18n^2p}\)
\(= \frac{5m^2p^2}{6n^2}\)
(s) \(\frac{mn^4pq}{m^2n^{-2}p^4}\)
\(= \frac{n^6q}{mp^3}\)
(t) \(\frac{46x^4y^2z^5}{69x^3y^2z^4}\)
\(= \frac{2xz}{3}\)
Notes
Sometimes the expressions that need to be cancelled are not a single term.
Examples and Your Turns
Example
Simplify$$\frac{5\left(x+y\right)}{10}$$
Your Turn
Simplify$$\frac{4}{2\left(a+b\right)}\\ \frac{5\left(x+y\right)}{7\left(x+y\right)}\\ \frac{\left(a+b\right)}{2\left(a+b\right)}\\ \frac{5\left(x-y\right)^{2}}{7\left(x-y\right)}\\ \frac{4\left(a-b\right)}{3\left(a-b\right)^{2}}\\ \frac{\left(x+y\right)\left(x-y\right)^{2}}{x\left(x-y\right)}\\ \frac{\left(a+b\right)\left(a-b\right)}{3\left(a+b\right)^{2}}\\ \frac{\left(x-y\right)}{\left(y-x\right)}\\ \frac{\left(1-x\right)}{\left(x-1\right)}\\ \frac{\left(3-2x\right)\left(2-x\right)}{\left(2x-3\right)\left(x+1\right)}\\ \frac{\left(3x-1\right)\left(3x-2\right)}{\left(2-3x\right)\left(3x+1\right)}$$
Further Practice
Simplify
(a) \(\frac{3(x + 2)}{3}\)
\( = x + 2 \)
(b) \(\frac{6(x + 2)}{(x + 2)}\)
\( = 6 \)
(c) \(\frac{(x + 2)(x - 1)}{(x - 1)(x + 3)}\)
\( = \frac{x + 2}{x + 3} \)
(d) \(\frac{4(x - 1)}{2}\)
\( = 2(x - 1) \text{ or } 2x - 2 \)
(e) \(\frac{x - 4}{2(x - 4)}\)
\( = \frac{1}{2} \)
(f) \(\frac{(x + 5)(2x - 1)}{3(2x - 1)}\)
\( = \frac{x + 5}{3} \)
(g) \(\frac{7(b + 2)}{14}\)
\( = \frac{b + 2}{2} \)
(h) \(\frac{2(x + 2)}{x(x + 2)}\)
\( = \frac{2}{x} \)
(i) \(\frac{(x + 6)^2}{3(x + 6)}\)
\( = \frac{x + 6}{3} \)
(j) \(\frac{2(n + 5)}{12}\)
\( = \frac{n + 5}{6} \)
(k) \(\frac{x(x - 5)^2}{3(x - 5)}\)
\( = \frac{x(x - 5)}{3} \)
(l) \(\frac{x^2(x + 2)}{x(x + 2)(x - 1)}\)
\( = \frac{x}{x - 1} \)
(m) \(\frac{10}{5(x + 2)}\)
\( = \frac{2}{x + 2} \)
(n) \(\frac{(x + 2)(x + 3)}{2(x + 2)^2}\)
\( = \frac{x + 3}{2(x + 2)} \)
(o) \(\frac{(x + 2)^2(x + 1)}{4(x + 2)}\)
\( = \frac{(x + 2)(x + 1)}{4} \)
(p) \(\frac{15}{5(3 - a)}\)
Notes
Most of the time we will have to factorise the numerator and/or denominator to be able to find common factors.
Examples and Your Turns
Example
Simplify$$\frac{2a+6}{4}$$
Your Turn
Simplify$$\frac{9b-3}{9}\\ \frac{2a^{2}+6a}{4}\\ \frac{9b^{2}-3b}{9}\\ \frac{2a^{2}+6a}{4a}\\ \frac{9b^{2}-3b}{9b}\\ \frac{3x+6y}{7x+14y}\\ \frac{44a-11b}{8a-2b}\\ \frac{3x-6y}{14y-7x}\\ \frac{11b-44a}{8a-2b}\\ \frac{r^{2}s+10rs}{3r+30}\\ \frac{5pq+25q}{10+2p^{2}}\\ \frac{p^{2}-9}{2p+6}\\ \frac{2x+10}{x^{2}-25}$$
Further Practice
Simplify
(a) \(\frac{2a - 2b}{b - a}\)
\( = -2 \)
(b) \(\frac{6x^2 - 3x}{1 - 2x}\)
\( = -3x \)
(c) \(\frac{m^2 - n^2}{n - m}\)
\( = -(m + n) \text{ or } -m - n \)
(d) \(\frac{3a - 3b}{6b - 6a}\)
\( = -\frac{1}{2} \)
(e) \(\frac{4x + 6}{4}\)
\( = \frac{2x + 3}{2} \text{ or } x + \frac{3}{2} \)
(f) \(\frac{3x + 6}{4 - x^2}\)
\( = \frac{3}{2 - x} \)
(g) \(\frac{a - b}{b - a}\)
\( = -1 \)
(h) \(\frac{12x - 6}{2x - x^2}\)
\( = \frac{6}{x} \)
(i) \(\frac{16 - x^2}{x^2 - 4x}\)
\( = \frac{-(4 + x)}{x} \text{ or } -\frac{4 + x}{x} \)
(j) \(\frac{a + b}{a - b}\)
\( = \frac{a + b}{a - b} \text{ (Cannot be simplified)} \)
(k) \(\frac{x^2 - 4}{x - 2}\)
\( = x + 2 \)
(l) \(\frac{x^2 - 4}{4 - x^2}\)
\( = -1 \)
(m) \(\frac{x - 2y}{4y - 2x}\)
\( = -\frac{1}{2} \)
(n) \(\frac{x^2 - 4}{x + 2}\)
\( = x - 2 \)
(o) \(\frac{5x^2 - 5y^2}{10xy - 10y^2}\)
\( = \frac{x + y}{2y} \)
(p) \(\frac{3m - 6n}{2n - m}\)
\( = -3 \)
(q) \(\frac{x^2 - 4}{2 - x}\)
\( = -(x + 2) \text{ or } -x - 2 \)
(r) \(\frac{2d^2 - 2a^2}{a^2 - ad}\)
\( = \frac{2(d + a)}{-a} \text{ or } -\frac{2(d + a)}{a} \)
(s) \(\frac{3x - 3}{x - x^2}\)
\( = -\frac{3}{x} \)
(t) \(\frac{x + 3}{x^2 - 9}\)
\( = \frac{1}{x - 3} \)
(u) \(\frac{4x^2 - 8x}{x^2 - 4}\)
\( = \frac{4x}{x + 2} \)
(v) \(\frac{xy^2 - xy}{3 - 3y}\)
\( = -\frac{xy}{3} \)
(w) \(\frac{m^2 - n^2}{m + n}\)
\( = m - n \)
(x) \(\frac{3x^2 - 6x}{4 - x^2}\)
\( = \frac{3x}{-(x + 2)} \text{ or } -\frac{3x}{x + 2} \)
Example
Simplify$$\frac{x^{2}+6x+5}{x+5}$$
Your Turn
Simplify$$\frac{x-2}{x^{2}-7x+10}\\ \frac{2x-12}{x^{2}-9x+18}\\ \frac{x^{2}+x-6}{3x+9}\\ \frac{x^{2}-2x-8}{x^{2}+3x+2}\\ \frac{x^{2}-8x+15}{2x^{2}-7x-15}$$
Further Practice
Simplify
(a) \(\frac{3a^2 - 6ab}{2a^2b - 4ab^2}\)
\( = \frac{3}{2b} \)
(b) \(\frac{x^3 - 2xy^2}{x^4 - 4x^2y^2 + 4y^4}\)
\( = \frac{x(x^2 - 2y^2)}{(x^2 - 2y^2)^2} = \frac{x}{x^2 - 2y^2} \)
(c) \(\frac{x^4 - 14x^2 - 51}{x^4 - 2x^2 - 15}\)
\( = \frac{(x^2 - 17)(x^2 + 3)}{(x^2 - 5)(x^2 + 3)} = \frac{x^2 - 17}{x^2 - 5} \)
(d) \(\frac{abx + bx^2}{acx + cx^2}\)
\( = \frac{b(a + x)}{c(a + x)} = \frac{b}{c} \)
(e) \(\frac{x^3y^2 - 27y^5}{x^3y^2 - 27y^5}\)
\( = 1 \)
(f) \(\frac{x^2 + xy - 2y^2}{x^3 - y^3}\)
\( = \frac{(x + 2y)(x - y)}{(x - y)(x^2 + xy + y^2)} = \frac{x + 2y}{x^2 + xy + y^2} \)
(g) \(\frac{ax}{a^2x^2 - ax}\)
\( = \frac{ax}{ax(ax - 1)} = \frac{1}{ax - 1} \)
(h) \(\frac{x^2 - 5x}{x^3 - 4x - 5}\)
\( = \frac{x(x - 5)}{(x - 5)(x^2 + 5x + 5)} = \frac{x}{x^2 + 5x + 5} \)
(i) \(\frac{2x^2 + 17x + 21}{3x^2 + 26x + 35}\)
\( = \frac{(2x + 3)(x + 7)}{(3x + 5)(x + 7)} = \frac{2x + 3}{3x + 5} \)
(j) \(\frac{15a^2b^2c}{100(a^3 - a^2b)}\)
\( = \frac{3c}{20(a - b)} \)
(k) \(\frac{3x^2 + 6x}{x^2 + 4x + 4}\)
\( = \frac{3x(x + 2)}{(x + 2)^2} = \frac{3x}{x + 2} \)
(l) \(\frac{a^2x^2 - 16a^2}{ax^3 + 9ax + 20a}\)
\( = \frac{a^2(x - 4)(x + 4)}{ax(x^2 + 9) + 20a} \quad \text{Mistake in Question/No further simple factorisation} \)
*(Assuming denominator was meant to be $ax^2 + 9ax + 20a$:* \( = \frac{a^2(x - 4)(x + 4)}{a(x^2 + 9x + 20)} = \frac{a(x - 4)}{x + 5} \text{)}*
*(Assuming denominator was meant to be $ax^2 + 9ax + 20a$:* \( = \frac{a^2(x - 4)(x + 4)}{a(x^2 + 9x + 20)} = \frac{a(x - 4)}{x + 5} \text{)}*
(m) \(\frac{4x^2 - 9y^2}{4x^2 + 6xy}\)
\( = \frac{(2x - 3y)(2x + 3y)}{2x(2x + 3y)} = \frac{2x - 3y}{2x} \)
(n) \(\frac{5a^3b + 10a^2b^2}{3a^2b^3 + 6ab^3}\)
\( = \frac{5a^2b(a + 2b)}{3ab^3(a + 2b)} = \frac{5a}{3b^2} \)
(o) \(\frac{3x^2 + 23x + 14}{3x^2 + 41x + 26}\)
\( = \frac{(3x + 2)(x + 7)}{(3x + 2)(x + 13)} = \frac{x + 7}{x + 13} \)
(p) \(\frac{20(x^3 - y^3)}{5x^2 + 5xy + 5y^2}\)
\( = \frac{20(x - y)(x^2 + xy + y^2)}{5(x^2 + xy + y^2)} = 4(x - y) \)
(q) \(\frac{x^3y + 2x^2y + 4xy}{x^3 - 8}\)
\( = \frac{xy(x^2 + 2x + 4)}{(x - 2)(x^2 + 2x + 4)} = \frac{xy}{x - 2} \)
(r) \(\frac{27a + a^4}{18a - 6a^2 + 2a^3}\)
\( = \frac{a(27 + a^3)}{2a(9 - 3a + a^2)} = \frac{a(a + 3)(a^2 - 3a + 9)}{2a(a^2 - 3a + 9)} = \frac{a + 3}{2} \)
(s) \(\frac{x(x^2 - 3ax)}{a(4x^2 - 9x^3)}\)
\( = \frac{x^2(x - 3a)}{ax^2(4 - 9x)} = \frac{x - 3a}{a(4 - 9x)} \)
(t) \(\frac{3a^4 + 9a^3b + 6a^2b^2}{a^4 + a^3b - 2a^2b^2}\)
\( = \frac{3a^2(a^2 + 3ab + 2b^2)}{a^2(a^2 + ab - 2b^2)} = \frac{3(a + 2b)(a + b)}{(a + 2b)(a - b)} = \frac{3(a + b)}{a - b} \)
Notes
To multiply algebraic fractions, we multiply the numerators and multiply the denominators.
We then simplify the resulting fraction.
To divide algebraic fractions, we first take the reciprocal of the second fraction, then multiply.
Examples and Your Turns
Example
Simplify$$\frac{x}{2}\times\frac{3}{y}$$
Your Turn
Simplify$$\frac{5}{2a}\times\frac{3b}{4}$$
Example
Simplify$$\frac{12x}{y^{2}}\div\frac{54x^{2}}{7y}$$
Your Turn
Simplify$$\frac{3y^{2}}{x}\div\frac{6x^{2}y}{5}$$
Further Practice
Simplify
(a) \(\frac{2ab}{3cd} \times \frac{c^2d^3}{ab^2}\)
\( = \frac{2cd^2}{3b} \)
(b) \(\frac{26xk^2p^3}{58mp^4} \times \frac{2xk^3}{13pkm} \div \frac{2x^2k^4}{87m^2p^2}\)
\( = \frac{3xk}{p} \)
(c) \(\frac{12a^2bc}{8ab^3} \times \frac{24ab^2}{36bc^2}\)
\( = \frac{a^2}{b c} \)
(d) \(\frac{15b^2}{40c} \times \frac{27c^3}{81d^3} \div \frac{abc}{14d^3}\)
\( = \frac{7c}{4a} \)
(e) \(\frac{15xy z^3}{a^2b^3c} \times \frac{3a^3x}{5yz}\)
\( = \frac{9a x^2 z^2}{b^3 c} \)
(f) \(\frac{b^2}{3c} \times \frac{4c^2}{5d^3} \div \frac{16a^2b^2c^2}{15d^5}\)
\( = \frac{d^2}{4a^2 c^2} \)
(g) \(\frac{7a^2b^3}{9ax^2y} \times \frac{18x^2c}{15ac^4}\)
\( = \frac{14b^3}{15c^3y} \)
(h) \(\frac{8a x^2}{7by} \times \frac{49c y^2}{64d x^3}\)
\( = \frac{7c y}{8bd x} \)
(i) \(\frac{8m^2n^3}{5x^3yz} \times \frac{15xyz^2}{16mn^2}\)
\( = \frac{3n z}{2x^2} \)
(j) \(\frac{15abc}{100a^2bc} \times \frac{128x^3y^2z^2}{16xyz}\)
\( = \frac{6x^2yz}{5a} \)
(k) \(\frac{21k^2p^3}{13mn^2} \times \frac{39m^2}{28p^2k^3}\)
\( = \frac{9m p}{4n^2 k} \)
(l) \(\frac{45a^2b^3c^4}{27x^4y^3z} \times \frac{243x y^2 z^3}{180a^2b^3c}\)
\( = \frac{3c^3 z^2}{4x^3 y} \)
(m) \(\frac{6bc}{4b^2c} \times \frac{2c^2}{8a} \div \frac{6ac}{16b^2x}\)
\( = \frac{2b x}{a^2} \)
(n) \(\frac{104xyz k^2 p}{28x^2 y k p} \times \frac{56y^2 z^5 p}{26y z^2 k}\)
\( = 8 z^4 \)
(o) \(\frac{2x^2y}{3yz} \times \frac{5x^2z}{7xy^2} \div \frac{21x^2y^2}{40xy^3z}\)
\( = \frac{400 x^3 z}{441 y^2} \)
(p) \(\frac{m^3}{8n} \times \frac{36p^3q^2}{81m n} \div \frac{15mp x^5}{27n^2 x^3 y}\)
\( = \frac{m p^2 q^2 y}{10 x^2} \)
(q) \(\frac{7m^2p}{17x^2y} \times \frac{51yz}{21p^2n} \div \frac{m^2x^3}{pyz}\)
\( = \frac{z^2}{x^5 n} \)
(r) \(\frac{a^3}{b^3} \times \frac{x y^2}{ab} \div \frac{p b^2}{a x} \times \frac{a p}{b^2}\)
\( = \frac{a^4 x^2 y^2}{b^8} \)
Example
Simplify$$\frac{x^{2}-x}{2xy}\times\frac{4x^{2}}{x-1}$$
Your Turn
Simplify$$\frac{2x-10}{15}\times\frac{5}{x^{2}-5x}$$
Example
Simplify$$\frac{x^{2}+2x+1}{x+5}\div\frac{x+1}{x^{2}-25}$$
Your Turn
Simplify$$\frac{x^{2}-4x+3}{x^{2}+2x-63}\div\frac{x^{2}-7x+10}{6x+54}$$
Further Practice
Simplify
(a) \(\frac{14x^2 + 7x}{12x^2 + 24x} \div \frac{2x - 1}{2x + 4}\)
\( = \frac{7(2x + 1)}{6x} \)
(b) \(\frac{2x^2 + 13x + 15}{4x^2 - 9} \div \frac{2x^2 + 11x + 5}{4x^2 - 1}\)
\( = \frac{2x - 1}{2x - 3} \)
(c) \(\frac{a^2b^2 + 3ab}{4a^2 - 1} \div \frac{ab + 3}{2a + 1}\)
\( = \frac{a^2b}{2a - 1} \)
(d) \(\frac{x^2 - 14x - 15}{x^3 - 4x - 45} \div \frac{x^2 - 12x - 45}{x^3 - 6x - 27}\)
\( = \frac{(x + 1)(x - 3)}{x + 3} \)
(e) \(\frac{x^2 - 4a^2}{ax + 2a^2} \times \frac{2a}{x - 2a}\)
\( = \frac{2(x + 2a)}{a(x + 2a)} = \frac{2}{a} \)
(f) \(\frac{2x^2 - x - 1}{2x^3 + 5x + 2} \times \frac{4x^2 + 4x + 1}{16x^2 - 49}\)
\( = \frac{(x - 1)(2x + 1)^3}{(x^2 + 2x + 2)(4x - 7)(4x + 7)} \)
*(Assuming a typo in the denominator of the first fraction was meant to be $2x^3 + 5x^2 + 2x$:* \( = \frac{x - 1}{2x(4x - 7)} \text{)}*
*(Assuming a typo in the denominator of the first fraction was meant to be $2x^3 + 5x^2 + 2x$:* \( = \frac{x - 1}{2x(4x - 7)} \text{)}*
(g) \(\frac{a^2 - 121}{a^3 - 1} \div \frac{a + 11}{a^2 + a + 1}\)
\( = \frac{a - 11}{a - 1} \)
(h) \(\frac{b^2 - 27b}{2b^2 + 5b} \times \frac{4b^2 - 11b + 15}{2b^2 - 4b - 25}\)
\( = \frac{b(b - 27)}{b(2b + 5)} \times \frac{4b^2 - 11b + 15}{2b^2 - 4b - 25} \text{ (No simple simplification)} \)
*(Assuming a typo in the second fraction's denominator was $2b^2 - 4b - 30$:* \( = \frac{b - 27}{2(b - 5)} \text{)}*
*(Assuming a typo in the second fraction's denominator was $2b^2 - 4b - 30$:* \( = \frac{b - 27}{2(b - 5)} \text{)}*
(i) \(\frac{16x^2 - 9a^2}{x^2 - b^2} \times \frac{x - 2}{4x - 3a}\)
\( = \frac{(4x + 3a)(x - 2)}{x - b} \)
(j) \(\frac{x^3 - 6x^2 - 36x}{x^2 - x^2} \div \frac{x^2 + 4x}{x^2 - 42}\)
\( = \frac{x(x^2 - 6x - 36)}{x^2(1 - 1)} \times \frac{x^2 - 42}{x(x + 4)} \)
*(This fraction is **undefined** due to $x^2 - x^2$ in the denominator.)*
*(Assuming the denominator was $x^3 - x^2$:* \( = \frac{(x - 12)(x - 6)(x + 6)}{x(x - 1)(x + 4)} \text{)}*
*(This fraction is **undefined** due to $x^2 - x^2$ in the denominator.)*
*(Assuming the denominator was $x^3 - x^2$:* \( = \frac{(x - 12)(x - 6)(x + 6)}{x(x - 1)(x + 4)} \text{)}*
(k) \(\frac{25a^2 - b^2}{9a^2x^2 - 4x^2} \times \frac{x(3a + 2)}{5a + b}\)
\( = \frac{5a - b}{x(3a - 2)} \)
(l) \(\frac{64p^2q^2 - z^4}{x^2 - 4} \div \frac{(x - z)^2}{8pq + z^2(x + 2)}\)
\( = \frac{(8pq + z^2)}{(x - 2)} \times \frac{(8pq - z^2)(x + 2)}{(x - z)^2} \text{ (No simple simplification)} \)
(m) \(\frac{x^2 + 5x + 6}{x^3 - 1} \times \frac{x^2 - 2x - 3}{x^2 - 9}\)
\( = \frac{x + 2}{x^2 + x + 1} \)
(n) \(\frac{x^2 - 18x + 80}{x^2 - 5x - 50} \times \frac{x^2 - 15x + 56}{x^2 - 15x + 56} \times \frac{x - 5}{x - 1}\)
\( = \frac{x - 8}{x - 10} \)
(o) \(\frac{x^2 + 3x + 2}{x^2 + 9x + 20} \times \frac{x^2 + 7x + 12}{x^2 + 5x + 6}\)
\( = \frac{x + 1}{x + 5} \)
(p) \(\frac{x^2 - 8x - 9}{x^2 - 17x + 72} \div \frac{x^2 - 25}{x^2 - 1} \times \frac{x^2 + 4x - 5}{x - 9x - 8}\)
\( = \frac{(x + 1)(x - 1)}{x - 5} \times \frac{x - 5}{x - 9x - 8} \text{ (Last term simplifies poorly)} \)
*(Assuming the last term's denominator was $x^2 - 9x + 8$:* \( = \frac{x + 1}{x - 8} \text{)}*
*(Assuming the last term's denominator was $x^2 - 9x + 8$:* \( = \frac{x + 1}{x - 8} \text{)}*
(q) \(\frac{2x^2 + 5x + 2}{x^2 - 4} \div \frac{2x^2 + 4x}{2x^2 + 9x + 4}\)
\( = \frac{(2x + 1)(x + 4)}{2x(x - 2)} \)
(r) \(\frac{4x^2 + x - 14}{6xy - 14y} \times \frac{4x^2}{x - 4} \div \frac{2x^2 + 4x}{3x^2 - x - 14}\)
\( = \frac{2x(2x + 7)}{y(x + 2)} \)
(s) \(\frac{x^2 + x - 2}{x^2 - x - 20} \div \frac{x^2 + 5x + 4}{x^2 - x} \times \frac{x - 2}{x + 3} \div \frac{x^2 + x + 3}{x^2 - 2x - 15}\)
\( = \frac{x(x - 2)}{x + 4} \)
(t) \(\frac{x^2 - 4x + 16}{x^2 + 12x + 64} \times \frac{x^2 - 64}{x^3 - 64} \div \frac{x^2 - 16x + 64}{x^2 + 4x + 16}\)
\( = \frac{x + 8}{(x + 8)(x - 8)} \times \frac{(x - 8)(x + 8)}{(x - 4)(x^2 + 4x + 16)} \times \frac{x^2 + 4x + 16}{(x - 8)^2} = \frac{1}{x - 8} \)
(u) \(\frac{4x^2 - 16x + 15}{2x^2 + 3x + 1} \times \frac{x^2 - 6x + 7}{2x^2 - 17x + 21} \times \frac{4x^2 - 1}{4x^2 - 20x + 25}\)
\( = \frac{(2x - 5)(2x - 3)}{(2x + 1)(x + 1)} \times \frac{x^2 - 6x + 7}{(2x - 3)(x - 7)} \times \frac{(2x - 1)(2x + 1)}{(2x - 5)^2} = \frac{(2x - 1)(x^2 - 6x + 7)}{(x + 1)(x - 7)(2x - 5)} \)
(v) \(\frac{x^4 - 8x}{x^4 - 4x - 5} \times \frac{x^2 + 2x + 4}{x^3 - x^2} \div \frac{x + 2x}{x - 5}\)
\( = \frac{x(x - 2)(x^2 + 2x + 4)}{(x - 5)(x^3 + x^2 + x + 1)} \times \frac{x^2 + 2x + 4}{x^2(x - 1)} \times \frac{x - 5}{3x} \text{ (Denominator of first fraction is prime)} \)
*(Assuming the denominator of the first fraction was $x^4 - 4x^2 - 5$:* \( = \frac{x^2 + 2x + 4}{x^2(x + 1)} \text{)}*
*(Assuming the denominator of the first fraction was $x^4 - 4x^2 - 5$:* \( = \frac{x^2 + 2x + 4}{x^2(x + 1)} \text{)}*
(w) \(\frac{(a + b)^2 - c^2}{a^2 + ab - ac} \div \frac{a}{(a + c)^2 - b^2} \times \frac{(a - b)^2 - c^2}{a b - b^2 - b c}\)
\( = \frac{(a + b + c)(a + c + b)}{b} \text{ or } \frac{(a + b + c)^2}{b} \)
(x) \(\frac{a^2 + 2ab + c^2 - b^2}{a^2 - b^2 - c^2 - 2bc} \times \frac{b^2 - 2bc + c^2 - a^2}{b^2 - 2b c + c^2 - a^2}\)
\( = \frac{a + b + c}{a - b - c} \)
Notes
To add and subtract algebraic fractions we must first find a common denominator and manipulate all fractions to have this denominator.
Then we can add or subtract the numerators over the common denominator.
Finally, we look to see if we can simplify our answer.
Examples and Your Turns
Example
Calculate$$\frac{4x-1}{3}+\frac{2x+1}{2}$$
Your Turn
Calculate$$\frac{x+4}{5}+\frac{2x-1}{3}$$
Your Turn
Calculate $$\frac{x+1}{2}-\frac{x-1}{3}$$
Your Turn
Calculate $$\frac{2x+3}{3}-\frac{3x-5}{5}$$
Your Turn
Calculate$$\frac{1}{2x}+\frac{2}{3x}\\ \frac{5}{3x}-\frac{2}{5x}\\ \frac{1}{x}+\frac{2}{x^{2}}\\ \frac{5}{x^{3}}-\frac{2}{x}\\ \frac{5}{y}-{2}{x}\\ \frac{4}{a}+\frac{1}{b}\\ \frac{y}{x^{2}}+\frac{7}{xy}+\frac{y}{x}\\ \frac{3}{y}+\frac{5}{xy}-\frac{x}{y^{2}}\\ 2-\frac{4}{x}\\ \frac{2}{x^{2}}+3\\ \frac{3}{x-1}-\frac{4}{x}\\ \frac{2}{x}+\frac{3}{x-1}\\ \frac{3}{x-1}-\frac{4}{x+2}\\ \frac{2}{x+1}+\frac{3}{x-3}$$
Further Practice
Calculate
(a) \(\frac{x}{2} + \frac{x}{3}\)
\( = \frac{5x}{6} \)
(b) \(\frac{1}{x + 1} + \frac{2}{x + 1}\)
\( = \frac{3}{x + 1} \)
(c) \(\frac{1}{x^2 - 4} + \frac{1}{x + 2}\)
\( = \frac{x - 1}{x^2 - 4} \text{ or } \frac{x - 1}{(x - 2)(x + 2)} \)
(d) \(\frac{2a}{5} - \frac{a}{10}\)
\( = \frac{3a}{10} \)
(e) \(\frac{3}{x - 2} - \frac{1}{x - 2}\)
\( = \frac{2}{x - 2} \)
(f) \(\frac{x}{x^2 - 9} - \frac{2}{x - 3}\)
\( = \frac{-x - 6}{x^2 - 9} \text{ or } \frac{-(x + 6)}{(x - 3)(x + 3)} \)
(g) \(\frac{3}{y} + \frac{5}{y}\)
\( = \frac{8}{y} \)
(h) \(\frac{5}{x - 3} + \frac{2}{3 - x}\)
\( = \frac{3}{x - 3} \)
(i) \(\frac{x + 1}{x^2 + 5x + 6} + \frac{x}{x + 3}\)
\( = \frac{x^2 + 3x + 1}{(x + 2)(x + 3)} \)
(j) \(\frac{7}{2m} - \frac{4}{2m}\)
\( = \frac{3}{2m} \)
(k) \(\frac{x}{a - b} - \frac{y}{b - a}\)
\( = \frac{x + y}{a - b} \)
(l) \(\frac{2x - 1}{x^2 - x - 6} - \frac{x}{x - 3}\)
\( = \frac{-x^2 - 1}{(x - 3)(x + 2)} \text{ or } \frac{-(x^2 + 1)}{x^2 - x - 6} \)
(m) \(\frac{x}{2} + \frac{y}{3}\)
\( = \frac{3x + 2y}{6} \)
(n) \(\frac{1}{x + 1} + \frac{1}{x + 2}\)
\( = \frac{2x + 3}{(x + 1)(x + 2)} \text{ or } \frac{2x + 3}{x^2 + 3x + 2} \)
(o) \(\frac{1}{x^2 + 2x + 1} + \frac{1}{x^2 - 1}\)
\( = \frac{2x}{(x + 1)^2 (x - 1)} \)
(p) \(\frac{3}{2a} - \frac{1}{b}\)
\( = \frac{3b - 2a}{2ab} \)
(q) \(\frac{2}{x - 3} - \frac{1}{x + 4}\)
\( = \frac{x + 11}{(x - 3)(x + 4)} \text{ or } \frac{x + 11}{x^2 + x - 12} \)
(r) \(\frac{x}{x^2 - 4} - \frac{x + 1}{x^2 + x - 2}\)
\( = \frac{-1}{(x - 2)(x + 1)} \text{ or } \frac{-1}{x^2 - x - 2} \)
(s) \(\frac{4}{x} + \frac{5}{2x}\)
\( = \frac{13}{2x} \)
(t) \(\frac{x}{x - 5} + \frac{x + 1}{x + 1}\)
\( = \frac{2x - 5}{x - 5} \)
(u) \(\frac{3x}{x^2 - xy - 2y^2} + \frac{y}{x^2 - 4y^2}\)
\( = \frac{3x^2 - 5xy + y^2}{(x - 2y)(x + y)(x + 2y)} \)
(v) \(\frac{3}{y^2} - \frac{2}{y}\)
\( = \frac{3 - 2y}{y^2} \)
(w) \(\frac{3}{2x + 1} + \frac{4}{x - 2}\)
\( = \frac{11x - 2}{(2x + 1)(x - 2)} \text{ or } \frac{11x - 2}{2x^2 - 3x - 2} \)
(x) \(\frac{x + y}{x^2 - 3xy + 2y^2} - \frac{x - y}{x^2 + xy - 2y^2}\)
\( = \frac{3y^2 - x^2}{(x - y)(x - 2y)(x + 2y)} \)
Notes
We can solve equations involving algebraic fractions.
The first step is usually to multiply through the whole equation by all denominators in order to ‘remove’ them and be left with an equation with no fractions.
Examples and Your Turns
Example
Solve$$\frac{x}{3}=6$$
Your Turn
Solve$$\frac{x}{5}=4\\ \frac{x}{3}+1=6\\ \frac{x}{5}-2=4\\ \frac{x+1}{3}=6\\ \frac{x-2}{5}=4\\ \frac{2x}{3}=6\\ \frac{2x}{5}=4\\ \frac{2x+7}{3}=6\\ \frac{2x-5}{5}=4\\ \frac{2x+7}{3}=6x-1\\ \frac{2x-5}{5}=4x+3$$
Your Turn
Solve$$\frac{3}{x}=6\\ \frac{5}{x}=4\\ \frac{3}{4x}=6\\ \frac{5}{3x}=4$$
Your Turn
Solve$$\frac{3x+1}{x-1}=-2\\ \frac{5x+1}{2-3x}=4$$
Your Turn
Solve$$\frac{4x-1}{5}+\frac{x+4}{2}=3$$
Your Turn
Solve$$\frac{2x+1}{4}-\frac{1-4x}{2}=\frac{3x+7}{6}$$
Your Turn
Solve$$\frac{6x-3}{2x+7}=\frac{3x-2}{x+5}$$
Your Turn
Solve$$\frac{x+4}{3x-8}=\frac{x+5}{3x-7}$$
Further Practice
Solve
(a) \(\frac{x + 4}{3x - 8} = \frac{x + 5}{3x - 7}\)
\( x = \frac{12}{5} \)
(b) \(\frac{x + 25}{x - 5} = \frac{2x + 75}{2x - 15}\)
\( x = 0 \)
(c) \(\frac{3x + 1}{3(x - 2)} = \frac{x - 2}{x - 1}\)
\( x = \frac{11}{3} \)
(d) \(\frac{x}{x + 2} + \frac{4}{x + 6} = 1\)
\( x = -3 \)
(e) \(\frac{7 - 5x}{1 + x} = \frac{11 - 15x}{1 + 3x}\)
\( x = -\frac{1}{2} \)
(f) \(\frac{6x + 7}{9x + 6} - \frac{1}{12} = \frac{5x - 5}{12x + 8}\)
\( x = -\frac{4}{3} \)
(g) \(\frac{3(7 + 6x)}{2 + 9x} = \frac{35 + 4x}{9 + 2x}\)
\( x = -\frac{7}{5} \)
(h) \(\frac{2x - 5}{5} + \frac{x - 3}{2x - 15} = \frac{4x - 3}{10} - 1\)
\( x = 5 \)
(i) \(\frac{6x + 13}{15} - \frac{3x + 5}{5x - 25} = \frac{2x}{5}\)
\( x = 10 \)
(j) \(\frac{4(x + 3)}{9} = \frac{8x + 37}{18} - \frac{7x - 29}{5x - 12}\)
\( x = 7 \)
(k) \(\frac{6x + 8}{2x + 1} - \frac{2x + 38}{x + 12} = 1\)
\( x = -7 \)
(l) \(\frac{(2x - 1)(3x + 8)}{6x(x + 4)} - 1 = 0\)
\( x = -\frac{8}{11} \)
(m) \(\frac{3x - 1}{2x - 1} - \frac{4x - 2}{3x - 1} = \frac{1}{6}\)
\( x = \frac{2}{3} \text{ or } x = 2 \)
(n) \(\frac{2x + 5}{5x + 3} - \frac{2x + 1}{5x + 2} = 0\)
\( x = -\frac{7}{5} \)
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.