8.1 – 8.5 Review
Section 8.1:
1. 𝑦 varies inversely as 𝑥. Write the appropriate inverse variation. Find 𝑦 for the given value of 𝑥.
1
𝑦 = −6 when 𝑥 = −8; Find y when 𝑥 =
2
2. 𝑦 varies jointly as 𝑥 and 𝑧. Write the appropriate joint-variation equation. Find 𝑦 for the given values of
𝑥 and 𝑧.
𝑦 = 100 when 𝑥 = −3 and 𝑧 = 14; Find 𝑦 when 𝑥 = 7 and 𝑧 = 9.
3. If 𝑥 varies directly as 𝑡 2 and inversely as 𝑦, and 𝑥 = 192 when 𝑡 = 8 and 𝑦 = 3, find 𝑦 when 𝑡 = 9 and 𝑥 =
486.
4. A bicycle’s pedal gear has 45 teeth and is rotating at 75 revolutions per minute. A chain links the pedal gear
to a rear-wheel gear that has 24 teeth.
a. How many revolutions per minute is the rear wheel gear moving?
b. How fast, in miles per hour, is the bicycle moving if the wheel’s diameter is 27 inches?
Section 8.2:
5. Determine whether each function is a rational function. If so, find the domain. If the function is not
rational, state why not.
a. 𝑓(𝑥) =
𝑥 4 +3𝑥 3 −2𝑥 2 +4𝑥−1
𝑥2
𝑥2
b. 𝑓(𝑥) = |𝑥|−2
6. Identify all asymptotes and holes in the graph of each rational function.
3𝑥+2
𝑥+7
a. 𝑚(𝑥) = 𝑥 2 −10
b. 𝑛(𝑥) = 𝑥 2 +4𝑥−21
7. Find the domain of each rational function. Identify all asymptotes and holes in the graph of each rational
function. Then graph.
𝑥 2 +2𝑥+1
𝑥 2 +4
a. 𝑦 = 𝑥 2 −3𝑥−4
b. 𝑦 = 4𝑥2 −1
Section 8.3:
8. Simplify each expression.
a.
c.
𝑥 2 +10𝑥+24
b.
𝑥 2 +2𝑥−24
14𝑥 3
27
∙
−9
−6𝑥 3
2𝑥
3
∙
5
d.
16𝑥 4 +112𝑥 3 +160𝑥 2
4𝑥 2 +8𝑥
𝑥 2 +8𝑥+16
𝑥 3 +10𝑥 2 +32𝑥+32
∙ 𝑥 2 + 8𝑥 + 16
e.
6𝑥 2 +30𝑥
𝑥 2 +6𝑥+5
÷
𝑥2 +11𝑥+24
𝑥2 −12𝑥+35
f.
5𝑥+40
2𝑥2 −14𝑥
𝑥 2 +4𝑥+4
𝑥 2 −𝑥−6
Section 8.4:
9. Write each expression as a single rational expression in simplest form.
a.
c.
e.
3
𝑥2
+
6
𝑥−5
7
b.
𝑥4
−
10
d.
𝑥+7
2𝑥
𝑥 2 −𝑥−6
−
𝑥+1
𝑥−3
+
𝑥+4
𝑥+2
f.
−4
𝑥 2 +2𝑥
+
𝑥+2
𝑥 2 −6𝑥−7
𝑥+2
3
𝑥−1
4
−
𝑥
𝑥+2
−
𝑥−2
𝑥 2 −𝑥−42
1
𝑥 2 +𝑥−2
Section 8.5:
10. Solve each equation.
a.
c.
𝑥−10
2𝑥+1
3
𝑥−1
=
4𝑥
b.
3𝑥+4
+4=
1
1−𝑥 2
d.
11. Solve each inequality.
a.
𝑥
𝑥−2
<2
b.
3
4
1
1
𝑥
2𝑥
− =
1
1+𝑐
𝑥+1
𝑥−1
−
1
2+𝑐
>2
=
1
4
12. Michael is training for a triathlon. He swims 0.6 miles, bicycles 15 miles, and runs 8 miles. Michael bicycles
about 9 times as fast as he swims, and he runs about 6 miles per hour faster than he swims.
a. Write a rational function, in terms of swimming speed, for the total time it takes Michael to
complete his workout.
b. Find the speeds at which Michael must swim, run, and bike to complete his workout in 1.5 hours.
13. A cab driver drove from the airport to a passenger’s home at an average speed of 55 miles per hour. He
returned to the airport along the same highway at an average speed of 45 miles per hour. What was the
cab driver’s average speed over the entire trip? (hint: the answer is not the average of 45 and 55)
14. Simplify
𝑥−𝑦
𝑥 −1 −𝑦 −1