In Exercises
ratsimpfirst -
ratsimplast , perform the indicated operations and simplify.
\(\frac {x^2-9}{x^2} \cdot \frac {3x}{x^2-x-6}\)
\(\frac {3(x+3)}{x(x+2)}\) , \(x \neq 3\)
\(\frac {t^2-2t}{t^2+1} \div (3t^2 - 2t - 8)\)
\(\frac {t}{(3t+4)(t^2+1)}\) , \(t \neq 2\)
\(\frac {4y-y^2}{2y+1} \div \frac {y^2-16}{2y^2-5y-3}\)
\(-\frac {y(y-3)}{y+4}\) , \(y \neq -\frac {1}{2}, 3, 4\)
\(\frac {x}{3x-1} - \frac {1-x}{3x-1} = \frac {\answer {2x-1}}{3x-1}\)
\(\frac {2}{w-1} - \frac {w^2+1}{w-1}\)
\(\frac {2-y}{3y} - \frac {1-y}{3y} + \frac {y^2-1}{3y} = \frac {\answer {y}}{3}\) , \(y \neq \answer {0}\)
\(b+ \frac {1}{b-3} - 2\)
\(\frac {b^2-5b+7}{b-3}\)
\(\frac {2x}{x-4} - \frac {1}{2x+1}\)
\(\frac {4x^2+x+4}{(x-4)(2x+1)}\)
\(\frac {m^2}{m^2-4} + \frac {1}{2-m} = \frac {\answer {m+1}}{m+2}\) , \(m \neq \answer {2}\)
\(\frac {\frac {2}{x} - 2}{x-1} = -\frac {2}{\answer {x}}\) , \(x \neq \answer {1}\)
\(\frac {\frac {3}{2-h} - \frac {3}{2}}{h}\)
\(\frac {3}{4-2h}\) , \(h \neq 0\)
\(\frac {\frac {1}{x+h} - \frac {1}{x}}{h}\)
\(-\frac {1}{x(x+h)}\) , \(h \neq 0\)
\(3w^{-1} - (3w)^{-1} = \frac {8}{\answer {3w}}\)
\(-2y^{-1} + 2(3-y)^{-2}\)
\(-\frac {2(y^2-7y+9)}{y(y-3)^2}\)
\(3(x-2)^{-1} - 3x(x-2)^{-2} = -\frac {6}{\answer {(x-2)^2}}\)
\(\frac {t^{-1} + t^{-2}}{t^{-3}}\)
\(\frac {2(3+h)^{-2} - 2(3)^{-2}}{h}\)
\(-\frac {2(h+6)}{9(h+3)^2}\) , \(h \neq 0\)
\(\frac {(7-x-h)^{-1} - (7-x)^{-1}}{h}\)
\(\frac {1}{(7-x)(7-x-h)}\) , \(h \neq 0\)
In Exercises
rateqnfirst -
rateqnlast , find all real solutions. Be sure to check for extraneous solutions.
\(\frac {x}{5x + 4} = 3\)
\(\frac {3y - 1}{y^{2} + 1} = 1\)
\(y = \answer {1}\) (smaller solution)
\(y = \answer {2}\) (larger solution)
\(\frac {1}{w + 3} + \frac {1}{w - 3} = \frac {w^{2} - 3}{w^{2} - 9}\)
\(\frac {2x + 17}{x + 1} = x + 5\)
\(x=\answer {-6}\) (smaller solution)
\(x=\answer {2}\) (larger solution)
\(\frac {t^{2} - 2t + 1}{t^{3} + t^{2} - 2t} = 1\)
\(\frac {-y^{3} + 4y}{y^{2} - 9} = 4y\)
\(y = 0, \pm 2\sqrt {2}\)
\(w + \sqrt {3} = \frac {3w - w^3}{w - \sqrt {3}}\)
\(w = \answer {-\sqrt {3}}\) (smaller solution.)
\(w=\answer {-1}\) (larger solution)
\(\frac {2}{x\sqrt {2} - 1} - 1 = \frac {3}{x \sqrt {2} + 1}\)
\(x = -\frac {3\sqrt {2}}{2}, \sqrt {2}\)
\(\frac {x^2}{(1 + x\sqrt {3})^2} = 3\)
\(x = \answer {-\frac {\sqrt {3}}{2}}\) (smaller solution)
\(x = \answer {-\frac {\sqrt {3}}{4}}\) (larger solution)
In Exercises
absratfirst -
absratlast , use Theorem
absvalequality along with the techniques in this section to find all real solutions.
\(\left |\frac {3n}{n-1} \right | = 3\)
\(n = \answer {\frac {1}{2}}\)
\(\left | \frac {2x}{x^2-1}\right | = 2\)
\(x = \frac {1 \pm \sqrt {5}}{2}, \frac {-1 \pm \sqrt {5}}{2}\)
\(\left | \frac {2t}{4-t^2}\right | = \left |\frac {2}{t-2}\right |\)
\(t = \answer {-1}\)
In Exercises
solveratcalcfirst -
solveratcalclast , find all real solutions and use a calculator to approximate your answers, rounded to two decimal
places.
\(2.41 = \frac {0.08}{4 \pi R^2}\)
\(R = \pm \sqrt {\frac {0.08}{9.64 \pi }} \approx \pm 0.05\)
\(\frac {x^2}{(2.31 -x)^2} = 0.04\)
\(x = -\frac {231}{400} \approx -0.58\) , \(x = \frac {77}{200} \approx 0.38\)
\(1 - \frac {6.75 \times 10^{16}}{c^2} = \frac {1}{4}\)
\(c = \pm \sqrt {\frac {4 \cdot 6.75 \times 10^{16}}{3}} = \pm 3.00 \times 10^{8}\)
NOTE: Challenge yourself - try to get the answer without using a calculator!
In Exercises
litrateqnfirst -
litrateqnlast , solve the given equation for the indicated variable.
Solve for
\(y\) :
\(\frac {1-2y}{y+3} = x\)
\(y = \frac {1 - 3x}{x+2}\) , \(y \neq -3\) , \(x \neq -2\)
Solve for
\(y\) :
\(x = 3 - \frac {2}{1-y}\)
\(y = \frac {x-1}{x-3}\) , \(y \neq 1\) , \(x \neq 3\)
Solve for
\(T_{2}\) :
\(\frac {V_{1}}{T_{1}} = \frac {V_{2}}{T_{2}}\)
NOTE: Recall that subscripts on variables have no intrinsic mathematical meaning; they’re just used to distinguish one
variable from another. In other words, treat quantities like ‘\(V_{1}\) ’ and ‘\(V_{2}\) ’ as two different variables as you would ‘\(x\) ’ and ‘\(y\) .
\(T_{2} = \frac {V_{2}}{T_{1}}{V_{1}}\) , \(T_{1} \neq 0, T_{2} \neq 0, V_{1} \neq 0\)
Solve for
\(t_{0}\) :
\(\frac {t_{0}}{1-t_{0}t_{1}} = 2\)
\(t_{0} = \frac {2}{2t_{1} + 1}\) , \(t_{1} \neq -\frac {1}{2}\)
Solve for
\(x\) :
\(\frac {1}{x - v_{r}} + \frac {1}{x + v_{r}} = 5\)
\(x = \frac {1 \pm \sqrt {25v_{r}^2+1}}{5}\) , \(x \neq \pm v_{r}\)
Solve for
\(R\) :
\(P = \frac {25R}{(R+4)^2}\)
\(R= \frac {-(8P-25) \pm \sqrt {(8P-25)^2 - 64P^2}}{2P} = \frac {(25-8P) \pm 5 \sqrt {25-16P}}{2P}\) , \(P \neq 0\) , \(R \neq -4\)
