3.4 Solve Rational Equations and Inequalities
Chapter
Chapter 3
Section
3.4
Lectures 5 Videos

ex Solve the inequality.

 \displaystyle \frac{1}{x + 1} - \frac{1}{2x + 3} \leq 0 

Solutions

x \in (-\infty, -2) \cup (-\frac{3}{2}, -1)

3.31mins
Rational Inequality ex1

ex Solve for x.

 \displaystyle \frac{x^2 -4}{(x -1)(x^3 -27)} \leq 0 

3.13mins
Rational Inequality ex2

ex Solve for x.

 \displaystyle \frac{(-x^2-3)(x^2-5x + 4)}{x^2 + 1} \geq 0 

Solutions

x \in [1, 4]

2.21mins
Rational Inequality ex3

ex Solve for x.

 \displaystyle \frac{1}{x} + \frac{1}{x -1} -\frac{1}{x + 5} \geq 0 

Solutions*

x \in [-5\sqrt{30}, -5]\cup (0, -5|\sqrt{30}]

4.47mins
Rational Inequality ex4
Solutions 51 Videos

Determine the x-intercept(s) for each function.

 \displaystyle y = \frac{x + 1}{x} 

0.17mins
Q1a

Determine the x-intercept(s) for each function.

 \displaystyle f(x) = \frac{x^2 + x - 12}{x^2 -3x + 5} 

0.21mins
Q1b

Determine the x-intercept(s) for each function.

 \displaystyle h(x) = \frac{2x-3}{5x + 1} 

0.12mins
Q1c

Determine the x-intercept(s) for each function.

 \displaystyle k(x) = \frac{x}{x^2 -3x+ 2} 

0.12mins
Q1d

Solve algebraically.

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

0.35mins
Q2a

Solve algebraically.

 \displaystyle \frac{1}{x^2 -2x - 7} =1 

0.51mins
Q2b

Solve algebraically.

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

0.44mins
Q2c

Solve algebraically.

 \displaystyle x - \frac{5}{x} = 4 

0.32mins
Q2d

Solve algebraically.

 \displaystyle \frac{1}{x} = \frac{x - 34}{2x^2} 

0.42mins
Q2e

Solve algebraically.

 \displaystyle \frac{x -3}{x -4} = \frac{x + 2}{x + 6} 

1.10mins
Q2f

Solve the inequality.

 \displaystyle \frac{4}{x - 3} < 1 

1.40mins
Q4a

Solve the inequality.

 \displaystyle \frac{7}{x+ 1} > 7 

1.08mins
Q4b

Solve the inequality.

 \displaystyle \frac{5}{x+ 4} \leq \frac{2}{x + 1} 

2.11mins
Q4c

Solve the inequality.

 \displaystyle \frac{(x-2)(x+1)^2}{(x - 4)(x+5)} \geq 0 

1.27mins
Q4d

Solve the inequality.

 \displaystyle \frac{x^2-16}{x^2 -4x - 5} > 0 

1.19mins
Q4e

Solve the inequality.

 \displaystyle \frac{x-2}{x} > \frac{x -4}{x - 6} 

2.02mins
Q4f

Solve each this inequality.

 \displaystyle \frac{x^2 + 9x + 14}{x^2 -6x + 5} > 0 

3.04mins
Q5a

Solve each this inequality.

 \displaystyle \frac{2x^2 + 5x -3}{x^2 + 8x + 16} < 0 

1.41mins
Q5b

Solve each this inequality.

 \displaystyle \frac{x^2 - 3x - 4}{x^2 + 11x + 30} \leq 0 

1.44mins
Q5c

Solve each this inequality.

 \displaystyle \frac{3x^2 - 8x + 4}{2x^2 - 9x - 5} \geq 0 

2.04mins
Q5d

Write a rational equation that cannot have x = 3 or x = -5 as a solution.

0.59mins
Q6

Solve \frac{x}{x + 1} < \frac{2x}{x - 2} by graphing the functions f(x) = \frac{x}{x + 1} and g(x) = \frac{2x}{x -2} with or without using technology. Determine the points of intersection and when f(x) < g(x).

5.22mins
Q7

Solve \displaystyle \frac{x}{x -3} > \frac{3x}{x + 5} by graphing two rational functions on a graphing device..

2.19mins
Q8

Solve.

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

0.30mins
Q9a

Solve.

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

1.43mins
Q9b

Solve.

 \displaystyle \frac{12}{x} + x = 8 

0.31mins
Q9c

Solve.

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

1.21mins
Q9d

Solve.

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

2.54mins
Q9e

Solve.

 \displaystyle \frac{7}{x - 2} - \frac{4}{x - 1}+ \frac{3}{x + 1} = 0 

2.24mins
Q9f

Solve the inequality.

 \displaystyle \frac{2}{x} + 3 > \frac{29}{x} 

1.20mins
Q10a

Solve the inequality.

 \displaystyle \frac{16}{x } -5 < \frac{1}{x} 

1.12mins
Q10b

Solve the inequality.

 \displaystyle \frac{5}{6x} + \frac{2}{3x} > \frac{3}{4} 

1.44mins
Q10c

Solve the inequality.

 \displaystyle 6 + \frac{30}{x - 1} < 7 

1.48mins
Q10d

The ratio of x + 2 to x - 5 is greater than \frac{3}{5}. Solve for x.

1.27mins
Q11

Compare the solutions to  \displaystyle \frac{2x -1}{x + 7} > \frac{x + 1}{x + 3} and \displaystyle \frac{2x - 1}{x+ 7} < \frac{x +1}{x + 3}

3.24mins
Q12

Compare the solutions to

i.  \displaystyle \frac{x + 1}{x-4} > \frac{x -3}{x + 5}

ii. and \displaystyle \frac{x -4}{x + 7} < \frac{x +5}{x - 3}

3.49mins
Q13

A number x is the harmonic mean of two numbers a and b is the mean of \frac{1}{a} and \frac{1}{b}.

• Write an equation to represent the harmonic mean of a and b.
0.24mins
Q14a

A number x is the harmonic mean of two numbers a and b is the mean of \frac{1}{a} and \frac{1}{b}.

• Determine the harmonic mean of 12 and 15.
0.40mins
Q14b

A number x is the harmonic mean of two numbers a and b is the mean of \dfrac{1}{a} and \dfrac{1}{b}.

• The harmonic mean of 6 and another number is 1.2. Determine the other number.
1.09mins
Q14c

The relationship between the object distance, d, and image distance, I, both in centimetres, for a camera with focal length 2.0cm is defined by the relation d = \dfrac{2.0I}{I - 2.0}. For what values of I is a d greater than 10.0 cm?

1.45mins
Q16

Consider the functions f(x) = \frac{1}{x} + 4 and g(x) = \frac{2}{x}. Graph f and g on the same grid.

• Determine the points of intersection of the two functions.
1.25mins
Q17a

Consider the functions f(x) = \frac{1}{x} + 4 and g(x) = \frac{2}{x}. Graph f and g on the same grid.

• i. Show where f(x) < g(x).
• ii. Solve the equation \frac{1}{x} + 4 = \frac{2}{x}.
1.47mins
Q17bcd

A rectangle has perimeter 64 cm and area 23 crn^2. Solve the following system of equations to find the rectangle’s width.

 \displaystyle l = \frac{23}{w} 

 \displaystyle l + 2 = 32 

1.17mins
Q18a

Solve the system of equations.

 \displaystyle \begin{array}{cccccc} &x^2 + y^2 = 1 \\ &xy = 0.5 \\ \end{array} 

2.49mins
Q18b

Use your knowledge of exponents to solve.

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

2.16mins
Q19a

Use your knowledge of exponents to solve.

 \displaystyle \frac{1}{2^x} = \frac{1}{x^2} 

1.33mins
Q19b

Determine the region(s) of the Cartesian plane for which

 \displaystyle y > \frac{1}{x^2} 

0.37mins
Q20a

Determine the region(s) of the Cartesian plane for which

 \displaystyle y\leq x^2 +4  and  \displaystyle y \geq \frac{1}{x^2 + 4} 

1.13mins
Q20b

Decompose each of the following into partial fractions.

 \displaystyle f(x) = \frac{5x + 7}{x^2 + 2x - 3} 

1.40mins
Q21a

Decompose each of the following into partial fractions.

 \displaystyle f(x) = \frac{7x +6}{x^2-x-6} 

 \displaystyle h(x) = \frac{6x^&2-14x-27}{(x + 2)(x -3)^2}