Solving Rational Equations Exercises

Subsection 1.5 Solving Rational Equations Exercises

Refer to \(P(x)=\frac{2}{4x-9}\) and \(S(x)=-\frac{x}{x-1}\text{.}\)
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1. What is the domain of \(P(x)\text{?}\)
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2. Graph \(P(x)\) so that the important features are visible, and label them.
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3. What is the domain of \(S(x)\text{?}\)
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4. Graph \(S(x)\) on the same set of axes as \(P(x)\text{.}\) Be sure that the important features are visible, and label them.
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5. Find the solutions to \(P(x)=S(x)\) using the graph on Desmos.
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6. Verify your solutions to \(P(x)=S(x)\) using algebra.
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7. Use your graph to solve \(P(x) \leq S(x)\text{.}\)
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Solve each equation.
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1. \(\displaystyle \frac{3}{k}=\frac{1}{k^2}+\frac{k+4}{k^2}\)
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2. \(\displaystyle 1=\frac{a-2}{4a}-\frac{1}{4a}\)
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3. \(\displaystyle \frac{3}{2m+8}=\frac{1}{m}-\frac{1}{2m^2+8m}\)
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4. \(\displaystyle \frac{1}{x^2-10x+24}+\frac{1}{x-4}=\frac{5}{x^2-10x+24}\)
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5. \(\displaystyle \frac{1}{2}=\frac{v+6}{2v-12}+\frac{3}{2}\)
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Supreme Scream is a ride in which passengers are dropped and experience the acceleration of gravity, before being rapidly brought to a complete stop. The ride reaches a height of 76.8 meters. An object falling under the force of gravity has height modeled by the equation \(H(t)=-4.9t^2 + vt + h\text{,}\) where \(H\) is measured in meters, \(v\) is the initial velocity in meters/second, and \(h\) is the initial height in meters.
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1. Keeping in mind that the ride drops passengers from a stop, write a function equation to model the height of passengers as a function of time since the ride started.
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2. If riders are allowed to free fall for 10 meters before the brakes are applied, how long (in seconds) is the free fall portion of the ride?
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3. If the speed of passengers on the ride is given by the equation \(S(t)=9.8t\) (meters/second), how fast are passengers traveling at the moment when the brakes are first applied?
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4. Suppose you are designing the next generation version of the ride, and you want riders to experience speeds of 80 miles/hour. How many meters must riders be allowed to free fall to reach this speed? (1 meter/second is the same as 2.24 miles/hour.)
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Refer to the function \(G(x)=\frac{3x^2-3}{(x-4)(x^2+1)}\text{.}\)
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1. Graph \(G(x)\) so that the important features are visible, and label them.
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2. Let \(H(x)=\frac{3x^2-3}{(x-4)(x^2-4)}\text{.}\) Graph \(H(x)\) so that the important features are visible, and label them.
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3. Explain why the graph of \(G\) has only one vertical asymptote, while \(H\) has three vertical asymptotes.
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4. What is the domain of \(G(x)\text{?}\)
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5. On which interval(s) is \(G(x)\) decreasing?
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6. What is the average rate of change of \(G(x)\) on the interval \(5 \leq x \leq 8\text{?}\)
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7. Let \(F(x)=G(x-2)\text{.}\) Graph \(F(x)\) and label its features.
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The population of Nairobi in Kenya was 2.14 million in 1999, and 3.14 million in 2009.
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1. Write a function equation \(P(t)\) for the number of people living in Nairobi \(t\) years after 1999.
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2. Rewrite \(P(t)\) using the natural base, \(e\text{.}\)
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3. What is the average number of additional residents added to Nairobi each year between 1999 and 2009?
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4. What will the population be in 2020, assuming it continues to grow at the same rate?
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5. When will the number of people living in Nairobi reach 5 million? First give an exact answer, and then give a decimal approximation.
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Solve the following equations.
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1. \(\displaystyle 2 \cdot 3^{x+4} = 162\)
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2. \(\displaystyle 64^{w-2}-\frac{1}{24}=\frac{1}{48}\)
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3. \(\displaystyle \ln (t+8) - \ln (t+13) = - \ln (2)\)
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