Lesson 11: Rational Equations, Formulas, and Shared Work Applications

Definitions and General Principles of Rational Equations

• A rational equation is defined as an equation that contains one or more rational expressions.

• A fundamental rule when working with rational equations is that division by zero is undefined; therefore, any value of a variable that causes any denominator in the equation to equal zero cannot be accepted as a valid solution ($\text{denominator} \neq 0$).

Step-by-Step Procedure for Solving Rational Equations

• Step 1: Factor all denominators. Completely factor all algebraic expressions located in the denominators across the entire equation.

• Step 2: Multiply by the Least Common Denominator (LCD). Multiply every term on both sides of the equation by the LCD to eliminate all fractional expressions.

• Step 3: Distribute and combine like terms. Expand any products generated by the multiplication and simplify the expressions on both sides of the equal sign.

• Step 4: Isolate the variable. Apply linear or quadratic equation solving strategies to solve for the target variable.

• Step 5: Verify all solutions. Check all candidate solutions against the original denominators to ensure they do not produce division by zero. Any solution that makes a denominator zero is an extraneous solution and must be discarded.

Worked Examples: Solving Rational Equations

• Example 1a: Solve the equation $-\frac{6}{x-5} - \frac{3}{x^2-9x+20} = -\frac{5}{x-4}$ and express the solution as a set.

  • Step 1: Factor the denominator $x^2 - 9x + 20$ into $(x - 5)(x - 4)$.

  • Step 2: Identify restrictions on the variable: $x \neq 5$ and $x \neq 4$

  • Step 3: Determine the LCD, which is $(x - 5)(x - 4)$.

  • Step 4: Multiply every term by $(x - 5)(x - 4)$:     $-6(x - 4) - 3 = -5(x - 5)$

  • Step 5: Distribute and combine like terms:     $-6x + 24 - 3 = -5x + 25$     $-6x + 21 = -5x + 25$

  • Step 6: Isolate the variable $x$:     $-x = 4$     $x = -4$

  • Step 7: Verify against domain restrictions: $x = -4$ does not make any original denominator zero ($-4 \neq 5$ and $-4 \neq 4$).

  • Final Solution Set: $\{-4\}$

• Example 1b: Solve the equation $\frac{3x-6}{x^2-4} - \frac{3}{x+2} = \frac{9}{x-2}$ and express the solution as a set.

  • Step 1: Factor the denominator $x^2 - 4$ into $(x - 2)(x + 2)$.

  • Step 2: Identify restrictions on the variable: $x \neq 2$ and $x \neq -2$

  • Step 3: Determine the LCD, which is $(x - 2)(x + 2)$.

  • Step 4: Multiply every term by $(x - 2)(x + 2)$:     $(3x - 6) - 3(x - 2) = 9(x + 2)$

  • Step 5: Distribute and combine like terms:     $3x - 6 - 3x + 6 = 9x + 18$     $0 = 9x + 18$

  • Step 6: Isolate the variable $x$:     $9x = -18$     $x = -2$

  • Step 7: Verify against domain restrictions: $x = -2$ causes division by zero in the denominators $(x^2 - 4)$ and $(x + 2)$, making it an extraneous solution.

  • Final Solution Set: $\emptyset$

• Example 1c: Solve the equation $\frac{17}{3} - \frac{3}{x} = -\frac{16}{x}$ and express the solution as a set.

  • Step 1: Identify restrictions on the variable: $x \neq 0$

  • Step 2: Determine the LCD, which is $3x$

  • Step 3: Multiply every term by $3x$:     $17x - 9 = -48$

  • Step 4: Isolate the variable $x$:     $17x = -39$     $x = -\frac{39}{17}$

  • Step 5: Verify against domain restrictions: $x = -\frac{39}{17}$ does not equal $0$

  • Final Solution Set: $\{-\frac{39}{17}\}$

• Example 1d: Solve the equation $\frac{6}{x+3} - \frac{5}{x-2} = \frac{-20}{x^2+x-6}$ and express the solution as a set.

  • Step 1: Factor the denominator $x^2 + x - 6$ into $(x + 3)(x - 2)$.

  • Step 2: Identify restrictions on the variable: $x \neq -3$ and $x \neq 2$

  • Step 3: Determine the LCD, which is $(x + 3)(x - 2)$.

  • Step 4: Multiply every term by $(x + 3)(x - 2)$:     $6(x - 2) - 5(x + 3) = -20$

  • Step 5: Distribute and combine like terms:     $6x - 12 - 5x - 15 = -20$     $x - 27 = -20$

  • Step 6: Isolate the variable $x$:     $x = 7$

  • Step 7: Verify against domain restrictions: $x = 7$ does not equal $-3$ or $2$

  • Final Solution Set: $\{7\}$

Step-by-Step Procedure for Solving Formulas with Rational Equations

• Step 1: Identify target variable. Clearly identify the variable you are attempting to isolate.

• Step 2: Eliminate fractions. Multiply all terms on both sides of the formula by the common denominator.

• Step 3: Distribute and combine like terms. Remove parentheses by distributing factors and simplifying expressions.

• Step 4: Isolate the specified variable using target techniques:

  • If the target variable appears in multiple terms, collect all terms containing that variable on one side of the equation and all terms without it on the opposite side. Then, factor out the target variable.

  • Add or subtract terms that do not contain the target variable to move them away from the target variable.

  • Divide or multiply by any remaining coefficients or factors attached to the target variable to complete the isolation.

Worked Examples: Solving Formulas for Specified Variables

• Example 2a: Solve the formula $v = \frac{2\pi r}{T}$ for $T$.

  • Step 1: Multiply both sides by $T$ to eliminate the denominator:     $v T = 2\pi r$

  • Step 2: Divide both sides by $v$ to isolate $T$:     $T = \frac{2\pi r}{v}$

• Example 2b: Solve the formula $\frac{qf + pf}{q} = p$ for $p$.

  • Step 1: Multiply both sides by $q$ to clear the denominator:     $qf + pf = pq$

  • Step 2: Group all terms containing the target variable $p$ on one side:     $qf = pq - pf$

  • Step 3: Factor out $p$ from the right side of the equation:     $qf = p(q - f)$

  • Step 4: Divide both sides by $(q - f)$ to isolate $p$:     $p = \frac{qf}{q - f}$

• Example 2c: Solve the formula $f = \frac{f_1 f_2}{f_1 + f_2}$ for $f_1$.

  • Step 1: Multiply both sides by $(f_1 + f_2)$ to clear the fraction:     $f(f_1 + f_2) = f_1 f_2$

  • Step 2: Distribute $f$ on the left side:     $f f_1 + f f_2 = f_1 f_2$

  • Step 3: Rearrange terms to group all terms containing $f_1$ on one side:     $f f_2 = f_1 f_2 - f f_1$

  • Step 4: Factor out $f_1$ from the right side:     $f f_2 = f_1(f_2 - f)$

  • Step 5: Divide both sides by $(f_2 - f)$ to isolate $f_1$:     $f_1 = \frac{f f_2}{f_2 - f}$

Shared Work Problems and Principles

• Concept of Shared Work: Shared work applications model scenarios where two individuals, machines, or objects perform a job together to complete it in less time than either working alone.

• Core Shared Work Equation:   $\text{Rate}_1 + \text{Rate}_2 = \text{Rate}_{\text{Together}}$

• Key Guidelines for Shared Work Problems:

  • 1. Establish Individual and Combined Rates:

  • If Person A completes a job in $2\,\text{hours}$, Person A's work rate is $\frac{1}{2}$ of the job per hour.

  • If Person B completes the same job in $3\,\text{hours}$, Person B's work rate is $\frac{1}{3}$ of the job per hour.

  • If the time required to complete the job together is unknown, represent it with variable $x$. The combined rate working together is $\frac{1}{x}$ of the job per hour.

  • Setting up the equation:       $\frac{1}{2} + \frac{1}{3} = \frac{1}{x}$

  • 2. Ensure Time Unit Consistency: Ensure all given times (hours, minutes, days) are converted to matching units before setting up equations.

  • 3. Reasonable Answer Check: Verify that the total calculated time working together is shorter than the time taken by any single individual working alone.

Worked Examples: Shared Work Applications

• Example 3: It takes a boy $90\,\text{minutes}$ to mow the lawn, but his sister can mow it in $1\,\text{hour}$. How long (in minutes) would it take them to mow the lawn if they worked together and each had their own lawn mower?

  • Step 1: Convert units so that time is consistently expressed in minutes.

  • Boy's time: $90\,\text{minutes}$

  • Sister's time: $1\,\text{hour} = 60\,\text{minutes}$

  • Step 2: Express individual rates per minute:

  • Boy's rate: $\frac{1}{90}$ of the lawn per minute

  • Sister's rate: $\frac{1}{60}$ of the lawn per minute

  • Combined rate together: $\frac{1}{x}$ of the lawn per minute, where $x$ is time in minutes

  • Step 3: Set up the shared work equation:     $\frac{1}{90} + \frac{1}{60} = \frac{1}{x}$

  • Step 4: Multiply by the LCD, which is $180x$:     $2x + 3x = 180$     $5x = 180$

  • Step 5: Solve for $x$:     $x = 36\,\text{minutes}$

  • Result: Working together, it takes them $36\,\text{minutes}$ to mow the lawn.

• Example 4: Working together, Bill and Tom painted a fence in $4\,\text{hours}$. If Tom had painted the same fence before by himself in $7\,\text{hours}$, how long would it take Bill (in hours) on his own? Round your answer to one decimal place.

  • Step 1: Express rates per hour:

  • Combined rate together: $\frac{1}{4}$ of the fence per hour

  • Tom's individual rate: $\frac{1}{7}$ of the fence per hour

  • Bill's individual rate: $\frac{1}{x}$ of the fence per hour, where $x$ is Bill's solo time in hours

  • Step 2: Set up the shared work equation:     $\frac{1}{x} + \frac{1}{7} = \frac{1}{4}$

  • Step 3: Multiply by the LCD, which is $28x$:     $28 + 4x = 7x$

  • Step 4: Solve for $x$:     $3x = 28$     $x = \frac{28}{3} \approx 9.3333...\,\text{hours}$

  • Step 5: Round to one decimal place as required:     $x \approx 9.3\,\text{hours}$

  • Result: Bill takes $9.3\,\text{hours}$ to paint the fence on his own.

• Example 5: Two pipes can be used to fill a swimming pool. When the first pipe is closed, the second pipe can fill the pool in $9\,\text{hours}$. When the second pipe is closed, the first pipe can fill the pool in $7\,\text{hours}$. How long (in hours) will it take to fill the pool if both pipes are open? Round your answer to two decimal places.

  • Step 1: Express individual filling rates per hour:

  • First pipe rate: $\frac{1}{7}$ of the pool per hour

  • Second pipe rate: $\frac{1}{9}$ of the pool per hour

  • Combined filling rate: $\frac{1}{x}$ of the pool per hour, where $x$ is total time in hours

  • Step 2: Set up the shared work equation:     $\frac{1}{7} + \frac{1}{9} = \frac{1}{x}$

  • Step 3: Multiply by the LCD, which is $63x$:     $9x + 7x = 63$     $16x = 63$

  • Step 4: Solve for $x$:     $x = \frac{63}{16} = 3.9375\,\text{hours}$

  • Step 5: Round to two decimal places as required:     $x \approx 3.94\,\text{hours}$

  • Result: With both pipes open, it will take $3.94\,\text{hours}$ to fill the pool.

• Example 6: On average, Mary can do her homework assignments in $100\,\text{minutes}$. It takes Frank about $2\,\text{hours}$ to complete a given assignment. How long (in minutes) will it take the two of them working together to complete an assignment? Round your answer to one decimal place.

  • Step 1: Convert units so that time is consistently expressed in minutes.

  • Mary's time: $100\,\text{minutes}$

  • Frank's time: $2\,\text{hours} = 120\,\text{minutes}$

  • Step 2: Express individual rates per minute:

  • Mary's rate: $\frac{1}{100}$ of the assignment per minute

  • Frank's rate: $\frac{1}{120}$ of the assignment per minute

  • Combined rate together: $\frac{1}{x}$ of the assignment per minute, where $x$ is total time in minutes

  • Step 3: Set up the shared work equation:     $\frac{1}{100} + \frac{1}{120} = \frac{1}{x}$

  • Step 4: Multiply by the LCD, which is $600x$:     $6x + 5x = 600$     $11x = 600$

  • Step 5: Solve for $x$:     $x = \frac{600}{11} \approx 54.5454...\,\text{minutes}$

  • Step 6: Round to one decimal place as required:     $x \approx 54.5\,\text{minutes}$

  • Result: Working together, it will take them $54.5\,\text{minutes}$ to complete an assignment.