Rational Equations

Rational Equations

I can solve rational algebraic equations.

PRE-ASSESSMENT SKILLS

How much do you know about rational equations?

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What is the value of x given the equation 𝑥 + 4 = −8?

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Find the product of  𝑎 ∙ 𝑏

 $𝑎=5\ 𝑎𝑛d\ 𝑏=𝑥+\frac{1}{5}$  .

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Simplify the equation

 $\frac{1}{x}+\frac{1}{2}=\frac{3}{2}\ .$  

Multiple Choice

Add the expression and simplify.

 $\frac{5}{18a^4b}+\frac{2}{27a^3b^2c}.$  

•  $\frac{15bc+4a}{54a^4b^2c}$  

•  $\frac{10bc+4a}{54a^4b^2c}$  

•  $\frac{15bc+3a}{54a^4b^2c}$  

•  $\frac{15𝑎𝑐+4𝑏}{54𝑎^4𝑏^2𝑐}$  

Multiple Choice

Subtract the expression

 $\frac{𝑡+3}{𝑡^2−6𝑡+9}\ by\ \ \frac{8𝑡−24}{𝑡^2−9}\ .$  

•  $\frac{−7𝑡+54𝑡^2−63}{𝑡^3−3𝑡^2−9𝑡+27}$  

•  $\frac{7𝑡^2+54𝑡−63}{𝑡^3−3𝑡^2−9𝑡+27}$  

•  $\frac{−7𝑡^2+54𝑡−63}{𝑡^3−3𝑡^2−9𝑡+27}$  

•  $\frac{−7𝑡^2−54𝑡−63}{𝑡^3−3𝑡^2−9𝑡+27}$  

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What is the value of a given the equation

 $\frac{3}{𝑎−2}=\frac{3}{4}\ .$  

Multiple Choice

Multiply the expression

 $\frac{4x}{3y}$  and  $\frac{3𝑥^2y^2}{10}.$  

•  $\frac{−2𝑥^3y}{5}$  

•  $\frac{2x^3y}{5}$  

•  $\frac{2x^2y}{5}$  

•  $\frac{-2x^2y}{5}$  

Multiple Choice

Simplify,  $\frac{\frac{c}{c^2-4}-\frac{c}{c-2}}{1+\frac{1}{c+2}}.$  

•  $\frac{c^2-c}{c^2+c-6}$  

•  $\frac{-c^2+c}{c^2+c-6}$  

•  $-\frac{c^2-c}{c^2+c-6}$  

•  $\frac{-c^2-c}{c^2+c-6}$  

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Perform the indicated operation,  $\frac{8}{b^2-4}\cdot\frac{b-2}{2b+4}\div\frac{4}{2\left(b+2\right)^2}\ .$  

Multiple Choice

Perform the indicated operation,  $\frac{1}{n-1}-\frac{1}{n^2-2n+1}+n\ .$  

•  $\frac{𝑛^4−2𝑛^3+2𝑛−2}{(𝑛−1)^2}$  

•  $\frac{𝑛^3−2𝑛^2−2𝑛+2}{(𝑛−1)^2}$  

•  $\frac{𝑛^3+2𝑛^2+2𝑛−2}{(𝑛−1)^2}$  

•  $\frac{𝑛^3−2𝑛^2+2𝑛−2}{(𝑛−1)^2}$  

Rational Equations

Example 1

Let's Practice 1

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Solve and check the solution.  $75=15+\frac{x}{3}\ .$  

Example 2

Extraneous root - is a root or solution derived from an equation that is not a solution of the original equation. Therefore, you must check all solutions in the original equation when you solve rational equations.

Example 3

Checking

Let's Practice 2

Multiple Choice

Solve the equation.  $\frac{9}{a+3}=1-\frac{a}{a-3}$  

•  $\frac{3}{2}$  

•  $\frac{9}{2}$  

•  $\frac{3}{a-3}$  

• No solution
  

Multiple Choice

Solve the equation.  $\frac{x}{x+1}+\frac{3x+5}{x^2+4x+3}=\frac{2}{x+3}\ .$  

•  $\frac{2x}{x+1}$  

•  $\frac{x}{x+3}$  

• No solution
  

•  $1$  

Solution and Answers

Let's Practice 1 and 2

FA 1.16

Solving Rational Equations

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1. Solve the equation.  $\frac{a}{2}-2=\frac{a}{3}$  

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2. Solve the equation.  $\frac{3}{b-2}=\frac{2}{b+2}$  

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3. Solve the equation.  $\frac{a+3}{a+7}-\frac{2a-14}{a^2+3a-28}=\frac{2-a}{4-a}$  

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4. Solve the equation.  $\frac{10}{2x+8}-\frac{7x+8}{x^2-16}=\frac{-8}{2x-8}$  

Multiple Choice

5. Solve for a in terms of b.  $\frac{1}{a}-\frac{2}{b+a}=\frac{1}{b-a}$  

•  $a=3b+1$  

•  $a=3b$ 
  

•  $a=\frac{3}{b}$  

•    $a=\frac{b}{3}$  
  

Solving Word Problems Involving Rational Equations

Number Problem

FA 1.17

Solving Number Problem

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1. The sum of the reciprocals of two consecutive integers is  $\frac{11}{30}\ .$  Find the integers. (Give the integers and separate them with a comma, no space in between)

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2. If the same number is added to both the numerator and the denominator of the fraction  $\frac{3}{5},$  the result is  $\frac{4}{5}\ .$  Find this number.

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3. The numerator is two less than the denominator. If one is subtracted from the numerator and added to the denominator, the value of the fraction is   $\frac{1}{3}.$  Find the original fraction. (Use "/" as a fraction bar, no space in between).

STUDY IN ADVANCE!

Motion and Work Problem pp. 165-166