HPC 2.6-2.8 Practice Makes Perfect
Review 2.6: Graph Rational Algebraic Functions
Accurately sketch the Rational Function. Identify and include all key features.
3x
x3  7 x 2  7 x  6
x2  2x
3x 2  11x  4
1. f ( x) 
4.
2.
3.
f
(
x
)

f
(
x
)

f
(
x
)

16  x 2
2 x 2  5 x  12
2 x 2  5 x  12
x 1
Review 2.7: Solve Rational Algebraic Equations
Solve the equation. Justify final solutions.
1.
4x
12
 1
x2
x
2.
1
x2
3

 2
x  2 x  5 x  3 x  10
3.
4x
x 1
2
 2

x  9 x  6x  9 x  3
2
Review 2.7: Using Rational Algebraic Fractions to Model Situations
Set up and solve the following problems:
1. How many gallons of a 12% salt solution should be added to 10 gallons of an 18% salt solution in order to
produce a solution whose salt content is between 14% and 16%?
2. On a 75-mile trip, Frank’s average rate of driving speed for the first 15 miles was 10 miles per hour less than
his average rate of driving speed for the rest of the trip. If the total driving time for the trip was 2 hours, find
the average rate of speed for the first 15 miles.
3. A 1000-liter tank contains 50 liters of a 25% brine solution. You add x liters of a 75% brine solution to the
tank. As the tank is filled, what happens to the rate at which the concentration of brine is increasing? What
percent does the concentration of brine appear to be approaching?
4. A page that is x inches wide and y inches high contains 30 square inches of print. The top and bottom
margins are 2 inches deep, and the margins on each side are 2 inches wide. What are the physical constraints
of this situation? Find the page size for which the least amount of paper will be used.
5. Let f(x) = x2 and g(x) = 3x + 10. Simplify the following expression:
6. If
𝑥2 +3𝑥𝑦−4𝑦2
𝑥2 −𝑦2
13
,
7
2
(𝑥+4)
=
𝑓(𝑥+3)−𝑔(𝑥+3)
.
(𝑥+3)+2
and xy, find the ratio of y to x.
2𝑥2 −12𝑥−31
7. Let 𝑓(𝑥) = 𝑥−3 and 𝑔(𝑥) =
. If ℎ(𝑥) = 𝑓(𝑥) + 𝑔(𝑥), find the zeros of ℎ(𝑥).
𝑥−3
8. Find all positive integral (integer) values of B for which the following equation has two distinct integral roots:
𝐵
𝑥 = 10−𝑥.
Review 2.8: Solve Algebraic Inequalities
Solve the inequality. Include a sign chart! Justify final solutions.
1.
x 1
0
x2  4
4.
1
2

x  2 x 1
2.
x 2  3x  10
0
x2  6 x  9
3. (2 x  1)2 x  4 x  2  0
x2  x  4
6.  x3  x 2  10 x  8  0
3
5.
x 1
0
Review 2.8: Solve Algebraic Inequalities
Set up and solve the following problems:
1. Pederson Electric Co. charges $25 per service call plus $18 per hour for home repair work. How long did
an electrician work if the charge was less than $100? Assume the electrician rounds off the time to the
nearest quarter hour.
2. The Grovenor Candy Co. finds that the cost of making a certain candy bar is $0.13 per bar. Fixed costs
amount to $2000 per week. If each bar sells for $0.35, find the minimum number of candy bars that will
earn the company a profit.
3. Flannery Cannery packs peaches in 0.5-L cylindrical cans.
a. Find the dimensions of the can if the surface is less than 900 cm2.
b. Find the volume of the can with the least possible surface area.
4. Write a paragraph that explains two ways to solve the inequality 3(x – 1) ≤ 2 (5x – 6). Include diagrams
and graphs to support your explanation.
5. The total electrical resistance R of two resistors connected in parallel with resistances R1 and R2 is given
by
1 1
1
. One resistor has a resistance of 2.3 ohms. Let x be the resistance of the second
 
R R1 R2
resistor.
a. Express the total resistance R as a function of x.
b. Find the resistance in the second resistor if the total resistance of the pair is at least 1.7 ohms.
6. Which of the following is/are true.
a. The solution set of x 2  25 is (5, )
x2
 2 can be solved by multiplying both sides by x  3 , resulting in the
x3
equivalent inequality x  2  2( x  3) .
x3
 0 have the same solution set.
c. ( x  3)( x  1)  0 and
x 1
b. The inequality
Running and Driving Time (hours)
d. None of these statements is true.
7. It’s vacation time. You drive 90 miles along a
scenic highway and then take a 5-mile run
along a hiking trail. Your driving rate is nine
times that of your running time. The graph
shows the total time you spend driving and
running, f(x), as a function of your running
rate. Use the rational function and its graph to
solve the following questions
a. Describe your running rate if you have
no more than a total of 3 hours for
driving and running. Use a rational
inequality to solve the problem. Then
explain how your solution can be
shown on the graph.
b. Describe the end behavior of the
graph as x → . What does this show you about the time driving and running as a function of
your running rate?
c. Describe the behavior of the graph as x → 0+. What does this show you about the time driving
and running as a function of your running rate?
d. Describe how to use the formula t 
function’s equation displayed.
s
and the problem’s verbal conditions to obtain the
v