TBIL-Precal Rational Equations (EQ6)

We will focus in this section on cases where the denominator is not a constant; that is, rational equations where there are variables in the denominator.

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With variables in the denominator, there will often be values that cause the denominator to be zero. This is a problem because division by zero is undefined. Thus, we need to be sure to exclude any values that would make those denominators equal to zero.

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Activity 1.6.3.

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Which value(s) should be excluded as possible solutions to the following rational equations? Select all that apply.

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(a)

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\frac{2}{x + 5} = \frac{x - 3}{x - 8} - 7

$- 7$
$- 5$
$2$
$3$
$8$

Answer.

B and E

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(b)

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\frac{x^{2} - 6 x + 8}{x^{2} - 4 x + 3} = 0

Answer.

B and D

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Activity 1.6.4.

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Consider the rational equation

5 = - \frac{6}{x - 2}

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(a)

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What value should be excluded as a possible solution?

$5$
$6$
$- 6$
$2$
$- 2$

Answer.

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(b)

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To solve, we begin by clearing out the fraction involved. What can we multiply each term by that will clear the fraction?

$x - 5$
$x - 6$
$x + 6$
$x - 2$
$x + 2$

Answer.

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(c)

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Multiply each term by the expression you chose and simplify. Which of the following linear equations does the rational equation simplify to?

$5 (x - 5) = - 6$
$5 (x - 6) = - 6$
$5 (x + 6) = - 6$
$5 (x - 2) = - 6$
$5 (x + 2) = - 6$

Answer.

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(d)

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Solve the linear equation. Check your answer using the original rational equation.

Answer.

x = \frac{4}{5}

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Activity 1.6.5.

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Consider the rational equation

\frac{4}{x + 1} = - \frac{2}{x + 6}

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(a)

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What values should be excluded as possible solutions?

$2$ and $4$
$1$ and $6$
$- 1$ and $- 6$
$1$ and $4$
$2$ and $6$

Answer.

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(b)

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To solve, we’ll once again begin by clearing out the fraction involved. Which of the following should we multiply each term by to clear out all of the fractions?

$x - 2$ and $x - 4$
$x - 1$ and $x - 6$
$x + 1$ and $x + 6$
$x - 1$ and $x - 4$
$x - 2$ and $x - 6$

Answer.

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(c)

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Multiply each term by the expressions you chose and simplify. Which of the following linear equations does the rational equation simplify to?

$4 (x + 1) = - 2 (x + 6)$
$4 (x + 6) = - 2 (x + 1)$
$4 (x + 1) (x + 6) = - 2 (x + 1) (x + 6)$
$4 (x + 1) = - 2 (- x - 6)$
$4 (x + 6) = - 2 (- x - 1)$

Answer.

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(d)

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Solve the linear equation. Check your answer using the original rational equation.

Answer.

x = - \frac{13}{3}

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Observation 1.6.6.

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In Activity 1.6.5, you may have noticed that the resulting linear equation looked like the result of cross-multiplying. This is no coincidence! Cross-multiplying is a method of clearing out fractions that works specifically when the equation is in proportional form:

\frac{a}{b} = \frac{c}{d} .

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Activity 1.6.7.

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Consider the rational equation

\frac{x}{x + 2} = - \frac{2}{x + 2} - \frac{2}{5}

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(a)

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What value(s) should be excluded as possible solutions?

Answer.

x = - 2

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(b)

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To solve, we’ll once again begin by clearing out the fraction involved. Which of the following should we multiply each term by to clear out all of the fractions?

$x + 2,$ $x + 2,$ and $5$
$x + 2$ and $5$
$x + 2$
$5$

Answer.

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(c)

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Multiply each term by the expressions you chose and simplify. You should end up with a linear equation.

Answer.

5 x = - 10 - 2 (x + 2)

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(d)

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Solve the linear equation. Check your answer using the original rational equation.

Answer.

No solution

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Observation 1.6.8.

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Activity 1.6.7 demonstrates why it is so important to determine excluded values and check our answers when solving rational equations. Just because a number is a solution to the linear equation we found, it doesn’t mean it is automatically a solution to the rational equation we started with.

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Activity 1.6.9.

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Consider the rational equation

\frac{2 x}{x - 1} - \frac{3}{x - 3} = \frac{x^{2} - 11 x + 18}{x^{2} - 4 x + 3}

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(a)

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What values should be excluded as possible solutions? Select all that apply.

Answer.

B and D

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(b)

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To solve, we’ll begin by clearing out any fractions involved. Which of the following should we multiply each term by to clear out all of the fractions?

$x - 1$
$x - 1$ and $x - 3$
$x - 1,$ $x - 3,$ and $x^{2} - 4 x + 3$
$x - 1$ and $x^{2} - 4 x + 3$
$x - 3$ and $x^{2} - 4 x + 3$

Answer.

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(c)

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Multiply each term by the expressions you chose and simplify. Notice that the result is a quadratic equation. Which of the following quadratic equations does the rational equation simplify to?