5.4 Solving Rational Equations
Chapter
Chapter 5
Section
5.4
Lectures 4 Videos

• Review of Linear: ax + b = 0
• Review of Quadratic: ax^2 + bx + c =0
8.47mins

Solving a Rational Equation ex1

Solve for x.

 \displaystyle \frac{3}{x} + \frac{5}{x + 2} = 2 

5.23mins
Solving a Rational Equation ex1

Solve each equation algebraically.

\displaystyle{\frac{2x}{2x+1}}=\displaystyle{\frac{5}{4-x}}

1.49mins
Q6a
Solutions 40 Videos

Are x=3 and x=-2 solutions to the equation \displaystyle{\frac{2}{x}}=\displaystyle{\frac{x-1}{3}}?

Explain how you know.

0.43mins
Q1

Solve each equation algebraically.

\displaystyle{\frac{x+3}{x-1}}=0

0.19mins
Q2a

Solve each equation algebraically.

\displaystyle{\frac{x+3}{x-1}}=2

0.27mins
Q2b

Solve each equation algebraically.

\displaystyle{\frac{x+3}{x-1}}=2x+1

0.55mins
Q2c

Solve each equation algebraically. Then verify your solution using graphing technology.

\displaystyle{\frac{3}{3x+2}}=\displaystyle{\frac{6}{5x}}

0.40mins
Q2d

For each rational equation, write a function whose zeros are the solutions.

\displaystyle{\frac{x-3}{x+3}}=2

0.39mins
Q3a

For each rational equation, write a function whose zeros are the solutions.

\displaystyle{\frac{3x-1}{x}}=\displaystyle{\frac{5}{2}}

0.31mins
Q3b

For each rational equation, write a function whose zeros are the solutions.

\displaystyle{\frac{x-1}{x}}=\displaystyle{\frac{x+1}{x+3}}

0.44mins
Q3c

For each rational equation, write a function whose zeros are the solutions.

\displaystyle{\frac{x-2}{x+3}}=\displaystyle{\frac{x-4}{x+5}}

0.59mins
Q3d

Solve algebraically. Verify your solution using a graphing calculator.

\displaystyle{\frac{x-3}{x+3}}=2

0.10mins
Q4a

Solve algebraically. Verify your solution using a graphing calculator.

\displaystyle{\frac{3x-1}{x}}=\displaystyle{\frac{5}{2}}

0.08mins
Q4b

Solve algebraically. Verify your solution using a graphing calculator.

\displaystyle{\frac{x-1}{x}}=\displaystyle{\frac{x+1}{x+3}}

0.07mins
Q4c

Solve algebraically. Verify your solution using a graphing calculator.

\displaystyle{\frac{x-2}{x+3}}=\displaystyle{\frac{x-4}{x+5}}

0.18mins
Q4d

Solve each equation algebraically.

\displaystyle{\frac{2}{x}}+\displaystyle{\frac{5}{3}}=\displaystyle{\frac{7}{x}}

0.32mins
Q5a

Solve each equation algebraically.

\displaystyle{\frac{10}{x+3}}+\displaystyle{\frac{10}{3}}=6

1.03mins
Q5b

Solve each equation algebraically.

\displaystyle{\frac{2x}{x-3}}=1-\displaystyle{\frac{6}{x-3}}

0.46mins
Q5c

Solve each equation algebraically.

\displaystyle{\frac{2}{x+1}}+\displaystyle{\frac{1}{x+1}}=3

0.21mins
Q5d

Solve each equation algebraically.

\displaystyle{\frac{2}{2x+1}}=\displaystyle{\frac{5}{4-x}}

0.46mins
Q5e

Solve each equation algebraically.

\displaystyle{\frac{5}{x-2}}=\displaystyle{\frac{4}{x+3}}

0.48mins
Q5f

Solve each equation algebraically.

\displaystyle{\frac{2x}{2x+1}}=\displaystyle{\frac{5}{4-x}}

1.49mins
Q6a

Solve each equation algebraically.

\displaystyle{\frac{3}{x}}+\displaystyle{\frac{4}{x+1}}=2

1.13mins
Q6b

Solve each equation algebraically.

\displaystyle{\frac{2x}{5}}=\displaystyle{\frac{x^2-5x}{5x}}

0.54mins
Q6c

Solve each equation algebraically.

x+\displaystyle{\frac{x}{x-2}}=0

0.31mins
Q6d

Solve each equation algebraically.

\displaystyle{\frac{1}{x+2}}+\displaystyle{\frac{24}{x+3}}=13

2.02mins
Q6e

Solve each equation algebraically.

\displaystyle{\frac{-2}{x-1}}=\displaystyle{\frac{x-8}{x+1}}

0.56mins
Q6f

Solve each equation using graphing technology. Round your answers to two decimal places, if necessary.

\displaystyle{\frac{2}{x+2}}=\displaystyle{\frac{3}{x+6}}

0.35mins
Q7a

Solve each equation using graphing technology. Round your answers to two decimal places, if necessary.

\displaystyle{\frac{2x-5}{x+10}}=\displaystyle{\frac{1}{x-6}} 

1.50mins
Q7b

Solve each equation using graphing technology. Round your answers to two decimal places, if necessary.

\displaystyle{\frac{1}{x-3}}=\displaystyle{\frac{x+2}{7x+14}}

0.41mins
Q7c

Solve each equation using graphing technology. Round your answers to two decimal places, if necessary.

\displaystyle{\frac{1}{x}}-\displaystyle{\frac{1}{45}}=\displaystyle{\frac{1}{2x-3}}

2.55mins
Q7d

Solve each equation using graphing technology. Round your answers to two decimal places, if necessary.

\displaystyle{\frac{2x+3}{3x-1}}=\displaystyle{\frac{x+2}{4}}

1.19mins
Q7e

Solve each equation using graphing technology. Round your answers to two decimal places, if necessary.

\displaystyle{\frac{1}{x}}=\displaystyle{\frac{2}{x}}+1+\displaystyle{\frac{1}{1-x}}

1.33mins
Q7f

Use algebra to solve \displaystyle{\frac{x+1}{x-2}}=\displaystyle{\frac{x+3}{x-4}}. Explain your steps.

1.41mins
Q8

The Greek mathematician Pythagoras is credited with the discovery of the Golden Rectangle. This is considered to be the rectangle with the dimensions that are the most Visually appealing. In a Golden Rectangle, the length and width are related by the proportion \frac{l}{w} = \frac{w}{l - w}. A billboard with a length of 15 m is going to be built. What must its width be to form a Golden Rectangle?

2.29mins
Q9

The Turtledove Chocolate factory has two chocolate machines. Machine A takes s minutes to fill a case with chocolates, and machine B takes s + 10 minutes to fill a case. Working together, the two machines take 15 min to fill a case. Approximately how long does each machine take to fill a case?

4.00mins
Q10

Tanya purchased a large box of comic books for $300. She gave 15 of the comic books to her brother and then sold the rest on an Internet website for \$330, making a profit of \\$1.50 on each one.

• How many comic books were in the box?
• What was the original price of each comic book?
4.02mins
Q11

Polluted water flows into a pond. The concentration of pollutant ,c, in the pond at time t minutes is modelled by the equation \displaystyle c(t) = 9 - 90 000(\frac{1}{10 000 + 3t})  where c is measured in kilograms per cubic metre.

a) When will the concentration of pollutant in the pond reach 6 kg/m^3?

b) What will happen to the concentration of pollutant over time?

2.37mins
Q12

Three employees work at a shipping warehouse. Tom can fill an order in s minutes. Paco can fill an order in s -2 minutes. Carl can fill an order in s + 1 minutes. When Tom and Paco work together, they take about 1 minute and 20 seconds to fill an order. When Paco and Carl work together, they take about 1 minute and 30 seconds to fill an order.

a) How long does each person take to fill an order?

b) How long would all three of them, working together, take to fill an order?

7.04mins
Q13

Solve  \displaystyle \frac{x^2-6x + 5}{x^2 -2x -3} = \frac{2 - 3x}{x^2 + 3x + 3} ,

correct to two decimal places.

5.01mins
Q15

Objects A and B move along a straight line. Their positions, 5, with respect to an origin, at 1‘ seconds, are modelled by the following functions:

you may use a graphing device for this Q

Object A: s_1(t) = \frac{7t}{t^2 + 1}

Object B: s_2(t) = t +\frac{5}{t + 2}

When are the objects at the same position?

2.44mins
Q16a

Objects A and B move along a straight line. Their positions, 5, with respect to an origin, at 1‘ seconds, are modelled by the following functions:

Object A: s_1(t) = \frac{7t}{t^2 + 1}

Object B: s_2(t) = t +\frac{5}{t + 2}

When is object A closer to the origin than object B?