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Section 1.5 Solving Rational Equations
F: Be able to produce a graph of a given rational function, indicating the vertical asymptotes, and
\(x\) - and
\(y\) -intercepts, if any.
F: Be able to solve equations involving rational functions.
An equation that involves rational expressions is a
rational equation .
Problem 1.5.1 .
Let
\(f(x)=\frac{x-2}{x+1}\)
Use Desmos to graph
\(f(x)\text{.}\) Sketch the graph below. Make sure the important features are visible and label them.
Figure 1.5.2. Blank coordinate plane with labeled axes and grid lines.
Solve
\(f(x)=4\) graphically.
Solve
\(f(x)=4\) algebraically.
What is the domain of
\(f(x)\text{?}\)
Solve
\(f(x) \geq 4\) graphically. How did you use the domain of
\(f(x)\) from
Item 4 to find your answer?
Problem 1.5.3 .
Let
\(g(x)=\frac{x}{x-4}\) and
\(h(x)=-\frac{1}{x+2}\text{.}\)
Use Desmos to graph
\(g(x)\) and
\(h(x)\text{.}\) Sketch the graph below. Make sure the important features are visible and label them.
Figure 1.5.4. Blank coordinate plane with labeled axes and grid lines.
Solve
\(g(x)=h(x)\) graphically.
Solve
\(g(x)=h(x)\) algebraically.
What is the value of
\(g(x)\) when
\(g(x)=h(x)\text{?}\)
What is the domain of
\(g(x)\text{?}\) What is the domain of
\(h(x)\text{?}\)
Solve
\(g(x)<h(x)\) graphically. How did you use the domain of
\(g(x)\) and
\(h(x)\) in
Item 5 to find your answer?
Problem 1.5.5 .
\(\displaystyle \frac{1}{x^2}+\frac{4}{x}=\frac{3}{x^2}\)
\(\displaystyle \frac{3}{2t} - \frac{2t}{t+1}=-2\)
\(\displaystyle \frac{y}{5y+5}=\frac{1}{y+2}+\frac{1}{y^2+3y+2}\)
Problem 1.5.6 .
\(\displaystyle \frac{2}{z+2}+2=\frac{7}{z+2}\)
\(\displaystyle \frac{1}{x-2} - \frac{1}{x^2-7x+10}=\frac{6}{x-2}\)
Definition 1.5.7 .
Sometimes in the process of solving a rational equation, you will obtain an
extraneous solution . An extraneous solution is a solution that appears to be valid, but does not satisfy the original equation.
Problem 1.5.8 .
Solve
\(\frac{1}{x+2}+\frac{1}{x-2}=\frac{4}{x^2-4}\)
Substitute your answer from
Item 1 back into the original equation? Does this answer satisfy the original equation?
Where in your solution process was this extraneous solution introduced?
Problem 1.5.9 .
Solve
\(\frac{x}{x-4}-\frac{4}{x+5}=\frac{36}{x^2+x-20}\text{.}\)
