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Grade 12

Identifying allowable values of rational equations

Lesson

A rational expression is one that has the form

$\frac{p\left(x\right)}{q\left(x\right)}$p(x)q(x),

where $p\left(x\right)$p(x) and $q\left(x\right)$q(x) are polynomials and $q\left(x\right)\ne0$q(x)0.

A rational equation is then an equation containing polynomials and at least one rational expression.

So, for example, the following are all rational equations:

$\frac{x}{x+1}=0$xx+1=0 $4=\frac{p^2-9}{2p-1}$4=p292p1 $5x^3-1=\frac{2x^3-3}{2x}$5x31=2x332x

 

Remember that a rational expression is undefined whenever the denominator of the expression is zero.

So an expression such as $\frac{1}{x}$1x is undefined for $x=0$x=0 and defined for every other value of $x$x.

More complicated expression such as $\frac{1}{x^2-3x+2}$1x23x+2 are also undefined for certain values of $x$x, but these values are less obvious. To find the values of $x$x for which the expression is undefined, we can construct an equation by setting the denominator to equal zero.

$x^2-3x+2$x23x+2 $=$= $0$0 (Setting the denominator to zero)
$\left(x-2\right)\left(x-1\right)$(x2)(x1) $=$= $0$0 (Factoring the expression)
$x$x $=$= $1,2$1,2 (Solving for each factor)

So we can conclude that the expression $\frac{1}{x^2-3x+2}$1x23x+2 is undefined for $x=1$x=1 and $x=2$x=2.

A rational equation is then undefined whenever any of the expressions in the equation are undefined.

 

Remember!

A rational expression is undefined whenever the denominator of the expression is zero.

A rational equation is then undefined whenever any expression in the equation is undefined.

 

Worked example

For what values of $x$x is the following rational equation undefined?

$\frac{x-3}{\left(x-3\right)\left(x+7\right)}=0$x3(x3)(x+7)=0

Think: A rational equation is undefined when the expressions of the equation are undefined. In this case, that happens when the denominator $\left(x-3\right)\left(x+7\right)$(x3)(x+7) is zero.

Do: To determine when the denominator is zero, we set it equal to zero and solve for $x$x:

$\left(x-3\right)\left(x+7\right)$(x3)(x+7) $=$= $0$0 (Setting the denominator to zero)
$x-3$x3 $=$= $0$0 (Solving when one factor is zero)
$x$x $=$= $3$3 (Adding $3$3 to both sides)
$x+7$x+7 $=$= $0$0 (Solving for the other factor)
$x$x $=$= $-7$7 (Subtracting $7$7 from both sides)
$x$x $=$= $-7,3$7,3 (Combining both solutions)

So we can conclude that the equation $\frac{x-3}{\left(x-3\right)\left(x+7\right)}=0$x3(x3)(x+7)=0 is undefined for $x=-7$x=7 and $x=3$x=3.

Reflect: The expression $\frac{x-3}{\left(x-3\right)\left(x+7\right)}$x3(x3)(x+7) has a common factor of $x-3$x3 between the numerator and denominator.

If we were to cancel this common factor, we would end up with the expression $\frac{1}{x+7}$1x+7. Notice that this expression is only undefined when $x=-7$x=7, and is defined when $x=3$x=3.

So the expressions $\frac{x-3}{\left(x-3\right)\left(x+7\right)}$x3(x3)(x+7) and $\frac{1}{x+7}$1x+7 are different, since the second one is defined for $x=3$x=3.

Now we can simplify a rational expression by cancelling common factors, but only over values for which it is defined.

That is, $\frac{x-3}{\left(x-3\right)\left(x+7\right)}=\frac{1}{x+7}$x3(x3)(x+7)=1x+7 for all real values of $x$x except for $x=-7$x=7 and $x=3$x=3.

 

Careful!

An algebraic expression is undefined whenever the denominator is zero before any factors have been cancelled.

 

Practice questions

Question 1

For what value of $w$w is the following rational equation undefined?

$\frac{4}{w}=0$4w=0

Question 2

For what values of $b$b is the following rational equation undefined?

$\frac{6}{b-2}-\frac{7b}{\left(b-5\right)\left(b-8\right)}=0$6b27b(b5)(b8)=0

  1. Write each answer on the same line, separated by commas.

question 3

For what values of $j$j is the following rational equation undefined?

$-\frac{4}{j}+\frac{7}{j+5}=\frac{24}{j^2+j}$4j+7j+5=24j2+j

  1. Write each answer on the same line, separated by commas.

Outcomes

12F.C.3.6

Solve simple rational equations in one variable algebraically, and verify solutions using technology

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