Practice Makes Perfect
Solve Rational Equations
In the following exercises, solve.
Example \(\PageIndex{37}\)
\(\frac{1}{a}+\frac{2}{5}=\frac{1}{2}\)
- Answer
-
10
Example \(\PageIndex{38}\)
\(\frac{5}{6}+\frac{3}{b}=\frac{1}{3}\)
Example \(\PageIndex{39}\)
\(\frac{5}{2}−\frac{1}{c}=\frac{3}{4}\)
- Answer
-
\(\frac{4}{7}\)
Example \(\PageIndex{40}\)
\(\frac{6}{3}−\frac{2}{d}=\frac{4}{9}\)
Example \(\PageIndex{41}\)
\(\frac{4}{5}+\frac{1}{4}=\frac{2}{v}\)
- Answer
-
\(\frac{40}{21}\)
Example \(\PageIndex{42}\)
\(\frac{3}{7}+\frac{2}{3}=\frac{1}{w}\)
Example \(\PageIndex{43}\)
\(\frac{7}{9}+\frac{1}{x}=\frac{2}{3}\)
- Answer
-
−9
Example \(\PageIndex{44}\)
\(\frac{3}{8}+\frac{2}{y}=\frac{1}{4}\)
Example \(\PageIndex{45}\)
\(1−\frac{2}{m}=\frac{8}{m^2}\)
- Answer
-
−2, 4
Example \(\PageIndex{46}\)
\(1+\frac{4}{n}=\frac{21}{n^2}\)
Example \(\PageIndex{47}\)
\(1+\frac{9}{p}=−\frac{20}{p^2}\)
- Answer
-
−5, −4
Example \(\PageIndex{48}\)
\(1−\frac{7}{q}=−\frac{6}{q^2}\)
Example \(\PageIndex{49}\)
\(\frac{1}{r+3}=\frac{4}{2r}\)
- Answer
-
−6
Example \(\PageIndex{50}\)
\(\frac{3}{t−6}=\frac{1}{t}\)
Example \(\PageIndex{51}\)
\(\frac{5}{3v−2}=\frac{7}{4v}\)
- Answer
-
14
Example \(\PageIndex{52}\)
\(\frac{8}{2w+1}=\frac{3}{w}\)
Example \(\PageIndex{53}\)
\(\frac{3}{x+4}+\frac{7}{x−4}=\frac{8}{x^2−16}\)
- Answer
-
\(-\frac{4}{5}\)
Example \(\PageIndex{54}\)
\(\frac{5}{y−9}+\frac{1}{y+9}=\frac{18}{y^2−81}\)
Example \(\PageIndex{55}\)
\(\frac{8}{z−10}+\frac{7}{z+10}=\frac{5}{z^2−100}\)
- Answer
-
−13
Example \(\PageIndex{56}\)
\(\frac{9}{a+11}+\frac{6}{a−11}=\frac{7}{a^2−121}\)
Example \(\PageIndex{57}\)
\(\frac{1}{q+4}−\frac{7}{q−2}=1\)
- Answer
-
no solution
Example \(\PageIndex{58}\)
\(\frac{3}{r+10}−\frac{4}{r−4}=1\)
Example \(\PageIndex{59}\)
\(\frac{1}{t+7}−\frac{5}{t−5}=1\)
- Answer
-
−5, −1
Example \(\PageIndex{60}\)
\(\frac{2}{s+7}−\frac{3}{s−3}=1\)
Example \(\PageIndex{61}\)
\(\frac{v−10}{v^2−5v+4}=\frac{3}{v−1}−\frac{6}{v−4}\)
- Answer
-
no solution
Example \(\PageIndex{62}\)
\(\frac{w+8}{w^2−11w+28}=\frac{5}{w−7}+\frac{2}{w−4}\)
Example \(\PageIndex{63}\)
\(\frac{x−10}{x^2+8x+12}=\frac{3}{x+2}+\frac{4}{x+6}\)
- Answer
-
no solution
Example \(\PageIndex{64}\)
\(\frac{y−3}{y^2−4y−5}=\frac{1}{y+1}+\frac{8}{y−5}\)
Example \(\PageIndex{65}\)
\(\frac{z}{16}+\frac{z+2}{4z}=\frac{1}{2z}\)
- Answer
-
−4
Example \(\PageIndex{66}\)
\(\frac{a}{9}+\frac{a+3}{3a}=\frac{1}{a}\)
Example \(\PageIndex{67}\)
\(\frac{b+3}{3b}+\frac{b}{24}=\frac{1}{b}\)
- Answer
-
−8
Example \(\PageIndex{68}\)
\(\frac{c+3}{12c}+\frac{c}{36}=\frac{1}{4c}\)
Example \(\PageIndex{69}\)
\(\frac{d}{d+3}=\frac{18}{d^2−9}+4\)
- Answer
-
2
Example \(\PageIndex{70}\)
\(\frac{m}{m+5}=\frac{50}{m^2−25}+6\)
Example \(\PageIndex{71}\)
\(\frac{n}{n+2}=\frac{8}{n^2−4}+3\)
- Answer
-
1
Example \(\PageIndex{72}\)
\(\frac{p}{p+7}=\frac{98}{p^2−49}+8\)
Example \(\PageIndex{73}\)
\(\frac{q}{3q−9}−\frac{3}{4q+12}=\frac{7q^2+6q+63}{24q^2−216}\)
- Answer
-
no solution
Example \(\PageIndex{74}\)
\(\frac{r}{3r−15}−\frac{1}{4r+20}=\frac{3r^2+17r+40}{12r^2−300}\)
Example \(\PageIndex{75}\)
\(\frac{s}{2s+6}−\frac{2}{5s+5}=\frac{5s^2−3s−7}{10s^2+40s+30}\)
- Answer
-
no solution
Example \(\PageIndex{76}\)
\(\frac{t}{6t−12}−\frac{5}{2t+10}=\frac{t^2−23t+70}{12t^2+36t−120}\)
Solve a Rational Equation for a Specific Variable
In the following exercises, solve.
Example \(\PageIndex{77}\)
\(\frac{C}{r}=2π\) for r
- Answer
-
\(r=\frac{C}{2π}\)
Example \(\PageIndex{78}\)
\(\frac{I}{r}=P\) for r
Example \(\PageIndex{79}\)
\(\frac{V}{h}=lw\) for h
- Answer
-
\(h=\frac{v}{lw}\)
Example \(\PageIndex{80}\)
\(\frac{2A}{b}=h\) for b
Example \(\PageIndex{81}\)
\(\frac{v+3}{w−1}=\frac{1}{2}\) for w
- Answer
-
w=2v+7
Example \(\PageIndex{82}\)
\(\frac{x+5}{2−y}=\frac{4}{3}\) for y
Example \(\PageIndex{83}\)
\(a=\frac{b+3}{c−2}\) for c
- Answer
-
\(c=\frac{b+3+2a}{a}\)
Example \(\PageIndex{84}\)
\(m=\frac{n}{2−n}\) for n
Example \(\PageIndex{85}\)
\(\frac{1}{p}+\frac{2}{q}=4\) for p
- Answer
-
\(p=\frac{q}{4q−2}\)
Example \(\PageIndex{86}\)
\(\frac{3}{s}+\frac{1}{t}=2\) for s
Example \(\PageIndex{87}\)
\(\frac{2}{v}+\frac{1}{5}=\frac{1}{2}\) for w
- Answer
-
\(w=\frac{15v}{10+v}\)
Example \(\PageIndex{88}\)
\(\frac{6}{x}+\frac{2}{3}=\frac{1}{y}\) for y
Example \(\PageIndex{89}\)
\(\frac{m+3}{n−2}=\frac{4}{5}\) for n
- Answer
-
\(n=\frac{5m+23}{m}\)
Example \(\PageIndex{90}\)
\(\frac{E}{c}=m^2\) for c
Example \(\PageIndex{91}\)
\(\frac{3}{x}−\frac{5}{y}=\frac{1}{4}\) for y
- Answer
-
\(y=\frac{20x}{12−x}\)
Example \(\PageIndex{92}\)
\(\frac{R}{T}=W\) for T
Example \(\PageIndex{93}\)
\(r=\frac{s}{3−t}\) for t
- Answer
-
\(t=\frac{3r−s}{r}\)
Example \(\PageIndex{94}\)
\(c=\frac{2}{a}+\frac{b}{5}\) for a
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?