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Wize High School Grade 12 Pre-Calculus Textbook > Rational Functions

Solving Rational Equations

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Solving Rational Equations

Rational equations are a type of equation containing at least one rational term, f(x)g(x)\boxed{\displaystyle\frac{f(x)}{g(x)}}, where f(x) & g(x)f(x)~\&~g(x) are continuous polynomials.


Rational equations can contain non-permissible values (NPV) that identify the restrictions of the domain. They identify the value of the variable that gives a 0 0~in the denominator.


Wize Tip
The NPV's are equivalent to the vertical asymptote and missing points/holes.

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Example

Solve 2x−1=5x+2\displaystyle\frac{2}{x-1}=\displaystyle\frac{5}{x+2}, stating any NPV's.


Step 1.
Identify any non-permissible values.

The NPV's are the vertical asymptotes.
Therefore, x≠−2,1x\neq{-2,1} otherwise there is a total of 00 in the denominator.


Step 2.
Find the lowest common denominator and multiply each term by it.

The lowest common denominator is (x−1)(x+2)(x-1)(x+2).

(2x−1=5x+2)×(x−1)(x+2)   →    2(x+2)=5(x−1)\Bigg(\displaystyle\frac{2}{x-1}=\displaystyle\frac{5}{x+2}\Bigg)\times(x-1)(x+2)~~~{\color{red}\rightarrow}~~~~2(x+2)=5(x-1)


Step 3.
Solve for ′x′.\green{'x'.}

2(x+2)=5(x−1)2x+4=5x−53x=9∴ x=3\begin{array}{} 2(x+2)&=&5(x-1)\\\\ 2x+4&=&5x-5\\\\ 3x&=&9\\\\ \therefore~x&=&3 \end{array}


Step 4.
Verify.

2x−1=5x+223−1=53+222=551=1✓\begin{array}{} \displaystyle\frac{2}{x-1}&=&\displaystyle\frac{5}{x+2}\\\\ \displaystyle\frac{2}{3-1}&=&\displaystyle\frac{5}{3+2}\\\\ \displaystyle\frac{2}{2}&=&\displaystyle\frac{5}{5}\\\\ 1&=&1&\checkmark \end{array}
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Example: Solving Rational Equations

Solve 3xx+5+1x−2=7x2+3x−10\displaystyle\frac{3x}{x+5}+\frac{1}{x-2}=\displaystyle\frac{7}{x^2+3x-10}, stating any NPV's.


Step 1.
Identify any non-permissible values.

The NPV's are the vertical asymptotes.

(x+5)=0→∴ x=−5(x−2)=0→∴ x=2x2+3x−10=(x+5)(x−2)=0→∴ x=−5,2\begin{array}{rccccccc} (x+5)&=&0&&\rightarrow&&\therefore~x&=&-5\\\\ (x-2)&=&0&&\rightarrow&&\therefore~x&=&2\\\\ x^2+3x-10=(x+5)(x-2)&=&0&&\rightarrow&&\therefore~x&=&-5, 2 \end{array}


Therefore, x≠−5,2x\neq{-5,2} otherwise there is a total of 00 in the denominator.


Step 2.
Find the lowest common denominator and multiply each term by it.

The lowest common denominator is (x+5)(x−2)(x+5)(x-2).

(3xx+5+1x−2=7x2+3x−10)×(x+5)(x−2)      →      3x(x−2)+(x+5)=7\Bigg(\displaystyle\frac{3x}{x+5}+\frac{1}{x-2}=\displaystyle\frac{7}{x^2+3x-10}\Bigg)\times(x+5)(x-2)~~~~~~{\color{red}\rightarrow}~~~~~~3x(x-2)+(x+5)=7


Step 3.
Solve for ′x′.\green{'x'.}

3x(x−2)+(x+5)=73x2−6x+x+5=73x2−5x−2=0(3x2−6x)+(x−2)=0(3x+1)(x−2)=0∴ x=−13,2\begin{array}{rcl} 3x(x-2)+(x+5)&=&7\\\\ 3x^2-6x+x+5&=&7\\\\ 3x^2-5x-2&=&0\\\\ (3x^2-6x)+(x-2)&=&0\\\\ (3x+1)(x-2)&=&0 \\\\\therefore~x&=&-\displaystyle\frac{1}{3}, 2 \end{array}

According to the NPV's, x≠2x\neq2.

Therefore, x=2x=2 is an extraneous solution.


Step 4.
Verify.

3xx+5+1x−2=7x2+3x−103(−13)(−13)+5+1(−13)−2=7(−13)2+3(−13)−10−314−37=−914−914=−914         ✓\begin{array}{rcl} \displaystyle\frac{3x}{x+5}+\frac{1}{x-2}&=&\displaystyle\frac{7}{x^2+3x-10}\\\\ \displaystyle\frac{3(-\frac{1}{3})}{(-\frac{1}{3})+5}+\frac{1}{(-\frac{1}{3})-2}&=&\displaystyle\frac{7}{(-\frac{1}{3})^2+3(-\frac{1}{3})-10} \\\\-\displaystyle\frac{3}{14}-\displaystyle\frac{3}{7}&=&-\displaystyle\frac{9}{14}\\\\ -\displaystyle\frac{9}{14}&=&-\displaystyle\frac{9}{14}~~~~~~~~~\checkmark \end{array}


Therefore, x=−13x=-\displaystyle\frac{1}{3} is the only solution.

Practice: Solving Rational Equations

True or false:

511 & x=4\displaystyle\frac{5}{11}~\&~x=4 are solutions to 3=2x+18x−33=\displaystyle\frac{2x+1}{8x-3}.

Practice: Solving Rational Equations

Given the equation 2x−3−34−x=2x−2x2−x−12\displaystyle\frac{2}{x-3}-\frac{3}{4-x}=\displaystyle\frac{2x-2}{x^2-x-12}, answer the following questions.



Solve for xx.


Practice: Solving Rational Equations

Let f(x)=x3+1x+1f(x)=\displaystyle\frac{x^3+1}{x+1}.


If f(x)=6x2+5x−7f(x)=6x^2+5x-7, then what is x?x?


Extra Practice
Solving Rational Equations
Practice: Solving Rational Equations
Let f(x)=4x2−252x+5f(x)=\dfrac{4x^2-25}{2x+5}.
If f(x)=4x2−2x+1f(x)=4x^2-2x+1, then what is x?x?
Solving Rational Equations
Practice: Solving Rational Equations
Let f(x)=x2−4x−2f(x)=\dfrac{x^2-4}{x-2}.
If f(x)=x2−5x−6f(x)=x^2-5x-6, then what is x?x?
Solving Rational Equations
Practice: Solving Rational Equations
Given the equation 4xx−5+3x+2=xx2−3x−10\dfrac{4x}{x-5}+\dfrac{3}{x+2}=\dfrac{x}{x^2-3x-10}, answer the following questions.
Solving Rational Equations
Practice: Solving Rational Equations
Given the equation 4x+1−1x+3=x+2x2+4x+3\dfrac{4}{x+1}-\dfrac{1}{x+3}=\dfrac{x+2}{x^2+4x+3}, answer the following questions.
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Practice: Solving Rational Equations
True or false:
8126\dfrac{81}{26} is a solution to 10=4x−13x+810=\dfrac{4x-1}{3x+8}.
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Practice: Solving Rational Equations
True or false:
177\dfrac{17}{7} is a solution to −5=2x+3x−4-5=\dfrac{2x+3}{x-4}.
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Solve for xx:
52x−1−73x+2=96x2+x−2\dfrac{5}{2x-1}-\dfrac{7}{3x+2}=\dfrac{9}{6x^2+x-2}
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Solve 2x−3x2−3=5x+1\dfrac{2x-3}{x^2-3}=\dfrac{5}{x+1}.