Solving Rational Equations

Show your work for each problem.

1.
Which of the following is a rational equation?
1. a)
2. b)
3. c)
4. d)
Answer:
is a rational equation because it has a variable () in the denominator. The other equations don't have variables in any denominator.
2.
Solve:
Answer: 3
Multiply both sides by : . Divide by 4: . Check: is true.
3.
What is the restricted value (value that makes a denominator zero) for ?
1. a)
2. b)
3. c)
4. d)
Answer:
The denominator is . Setting it to zero: , so . This value would make the fraction undefined.
4.
Solve by cross-multiplying:
Answer: 3
Cross multiply: . Divide by 8: .
5.
Solve:
Answer: 1/2
1. Combine the fractions on the left side (same denominator) 5/x
2. Rewrite the equation after combining 5/x = 10
3. Cross multiply to solve 5 = 10x
4. Solve for x 1/2
6.
Solve:
Answer: 5
Cross multiply: . Expand: . Solve: , so .
7.
What is the LCD (Least Common Denominator) for: ?
1. a)
2. b)
3. c)
4. d)
Answer:
The denominators are and . Since they share no common factors, the LCD is their product: .
8.
Solve:
Answer: -3
1. Cross multiply 4(x+1) = 2(x-1)
2. Expand both sides 4x + 4 = 2x - 2
3. Move all x terms to one side 2x = -6
4. Solve for x 3
9.
What is the result when you solve ?
1. a)
2. b)
3. c)Infinitely many solutions
4. d)No solution (extraneous)
Answer: No solution (extraneous)
Since denominators are equal, . But makes the denominator , which is undefined. So is extraneous and there is no solution.
10.
Solve:
Answer: 4
. Therefore .
11.
Solve:
Answer: 3
12.
Solve:
Answer: 8
LCD = . Multiply: . Expand: . Simplify: , so and . Check: doesn't make any denominator zero.
13.
Pipe A fills a tank in 6 hours. Pipe B fills it in 4 hours. How many hours to fill it together?
Answer: 2.4
1. What is Pipe A's rate (fraction of tank per hour)? 1/6
2. What is Pipe B's rate? 1/4
3. Write the equation for combined rate equaling 1/t 1/6 + 1/4 = 1/t
4. Find the LCD and add the fractions on the left 5/12
5. Solve for t 2.4
14.
Two resistors of 6 ohms and 12 ohms are connected in parallel. What is the total resistance using ?
1. a)3 ohms
2. b)9 ohms
3. c)18 ohms
4. d)4 ohms
Answer: 4 ohms
. Therefore ohms.
15.
A cyclist rides 30 km at 15 km/h, then returns at 10 km/h. What is the average speed for the entire trip (in km/h)?
Answer: 12
Time out = 30/15 = 2 hours. Time back = 30/10 = 3 hours. Total = 5 hours for 60 km. Average speed = 60/5 = 12 km/h. Note: NOT the average of 15 and 10!