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Section 8.5 Equations with Algebraic Fractions

Subsection 1. Solve equations graphically

If we can't solve an equation algebraically, we may be able use a graph to find at least an approximation for the solution.

Subsubsection Examples

Example 8.61.

Use a graph to solve the equation \(~2x^3+9x^2-8x+36=0\)

Solution

We graph the equation \(~y=2x^3+9x^2-8x-36~\) and look for the points where \(~y=0~\) (the \(x\)-intercepts).

cubic

From the graph, we estimate the solutions at \(~x=-4.5,~x=-2,\) and \(~x=2\text{.}\) By substituting each of these values into the original equation, you can verify that they are indeed solutions.

Example 8.62.

Use a graph to solve the equation \(~x^2+2x+3 = 15-2x\)

Solution

We graph the equations \(~y_1=x^2+2x+3~\) and \(~y_2=15-2x~\) and look for points on the two graphs where the coordinates are equal (intersection points).

parabola and line

From the graph, we see that the points with \(~x=-6~\) and \(~x=2~\) have the same \(y\)-coordinate on both graphs. In other words, \(~y_1=y_2~\) when \(~x=-6~\) or \(~x=2~\text{,}\) so \(~x=-6~\) and \(~x=2~\) are the solutions.

Subsubsection Exercises

Use a graph to solve the equation \(~2x^3+7x^2-7x-12=0\)

Answer
\(x = -4,~ -1,~ \dfrac{3}{2}\)

Use a graph to solve the equation \(~\dfrac{24}{x+4}=11+2x-x^2\)

Answer
\(x=-1,~4\)

Subsection 2. Choose the correct technique

We have learned a number of algebraic skills to deal with fractions. Each type of problem has a particular method.

Subsubsection Example

Example 8.65.

Choose the appropriate technique for each problem.

  1. Cross-multiply
  2. Multiply each term by the LCD
  3. Multiply top and bottom by the LCD
  4. Find building factors
  1. Combine \(~\dfrac{8}{x+2} + \dfrac{x}{x-3}\)
  2. Solve \(~\dfrac{8}{x+2} = \dfrac{x}{x-3}\)
  3. Solve \(~\dfrac{8}{x+2} + 1 = \dfrac{x}{x-3}\)
  4. Simplify \(~\dfrac{\dfrac{8}{x} + 1}{\dfrac{x}{x-3}+\dfrac{2}{x}}\)
Solution
  1. To add fractions, we find an LCD and build each fraction, so choice IV is correct.
  2. To solve a proportion, we can cross-multiply, so choice I is correct.
  3. To clear fractions from an equation, we multiply by the LCD, so choice II is correct.
  4. To simplify a complex fraction, we apply the fundamental pricnicple of fractions, so choice III is correct.

Subsubsection Exercises

Write the first step for the problem.

Solve \(~\dfrac{3}{x} + 3 = \dfrac{1}{x+3}\)

Answer
\(3(x+3)+3x(x+3)=x\)

Write the first step for the problem.

Combine \(~\dfrac{3}{x} + 3 - \dfrac{1}{x+3}\)

Answer
\(\dfrac{3(x+3)}{x(x+3)} + \dfrac{3x(x+3)}{x(x+3)} - \dfrac{x}{x(x+3)}\)

Write the first step for the problem.

Simplify \(~\dfrac{\dfrac{3}{x} + 1} {3- \dfrac{1}{x+3}}\)

Answer
\(\dfrac{3(x+3)-x(x+3)}{3x(x+3)-x}\)

Write the first step for the problem.

Solve \(~\dfrac{3}{x} = \dfrac{1}{x+3}\)

Answer
\(3(x+3)=x\)

Subsection 3. Solve quadratic equations

Once we have cleared the fractions from an equation, we may have a quadratic equation to solve. We can choose the easiest method to solve: factoring, extracting roots, or the quadratic formula.

Subsubsection Example

Example 8.70.

Solve each quadratic equation by the easiest method.

  1. \(\displaystyle 2x^2-2x=3\)
  2. \(\displaystyle (2x-1)^2=3\)
  3. \(\displaystyle 2x^2-x=3\)
Solution

  1. Because \(~2x^2-2x-3~\) does not factor, we use the quadratic formula.

    \begin{equation*} x = \dfrac{2 \pm \sqrt{(-2)^2-4(2)(-3)}}{2(2)} = \dfrac{2\pm \sqrt{28}}{4} = \dfrac{1 \pm \sqrt{7}}{2} \end{equation*}

  2. We use extraction of roots.

    \begin{align*} 2x-1 \amp = \pm \sqrt{3}\\ x \amp = \dfrac{1 \pm \sqrt{3}}{2} \end{align*}

  3. We write the equation in standard form and factor the left side.

    \begin{align*} 2x^2-x-3 \amp = 0\\ (2x-3)(x+1) \amp = 0\\ 2x-3=0~~~~x+1 \amp = 0\\ x = \dfrac{3}{2}~~~~x \amp = -1 \end{align*}

Subsubsection Exercise

Solve each equation by the easiest method.

  1. \(\displaystyle 3x^2+10x=8\)
  2. \(\displaystyle x^2+6x+9=8\)
  3. \(\displaystyle 81x^2-18x+1=0\)
  4. \(\displaystyle 9x^2+18x=27\)
Answer
  1. \(\displaystyle x=-4,~\dfrac{2}{3}\)
  2. \(\displaystyle x=-2\pm2\sqrt{2}\)
  3. \(\displaystyle x=\dfrac{1}{9},~\dfrac{1}{9}\)
  4. \(\displaystyle x=-3,~1\)