College Algebra
Chapter 4, section 7
Created by Lauren Atkinson
Mary Stangler Center for Academic Success
This review is meant to highlight basic concepts from Chapter 4. It does not cover all concepts presented by your instructor. Refer back
to your notes, handouts, the book, MyMathLab, etc. for further prepare for your exam.
4.7: More equations and Inequalities
• This section covers:
– Rational equations
– Polynomial/Rational Inequalities
– Direct and Inverse Variation
Rational Equations
• These can now equal a constant 𝑘 or another
rational function entirely. We then have to
solve for 𝑥.
𝑃(𝑥)
𝑄(𝑥)
= 𝑘 or
𝑃(𝑥)
𝑄(𝑥)
=
𝑅(𝑥)
𝑆(𝑥)
Examples:
Solve:
4
3
=
𝑥−2 𝑥−1
1
1
=
𝑥2 − 2 𝑥
1
2
−1
−
= 2
𝑥 − 2 𝑥 − 3 𝑥 − 5𝑥 + 6
𝑥 = −2 (checked and confirmed)
𝑥 = −1, 2 (checked and confirmed)
𝑁𝑂 𝑆𝑂𝐿𝑈𝑇𝐼𝑂𝑁 (𝑥 = 2, when
plugged back in gives us division by
zero which is no solution)
IT IS EXTREMEMLY IMPORTANT THAT WE
CHECK OUR ANSWERS! Just because we find
an x, does not mean that x is actually an
answer.
Polynomial and Rational Inequalities
We follow a similar “step-by-step” method for
solving both polynomial and rational
inequalities:
1. Solve 𝑃(𝑥) = 0 or in
𝑃(𝑥)
solve
𝑄(𝑥)
both 𝑃(𝑥) = 0 and 𝑄(𝑥) = 0
[these are the “critical points”]
2. Use those critical points to form intervals with both −∞ and
∞ included
3. Use a table to solve the inequality
interval
Test point
𝑓(test point)
Fill these in based on the problem
Is 𝒇(test point) positive or negative?
This is best explained with examples:
Solve the polynomial inequality:
3
𝑥 −𝑥 >0
interval
−∞, −1
−1,0
0,1
(1, ∞)
Critical Points:
𝑥3 − 𝑥 = 0
𝑥 𝑥2 − 1 = 0
𝑥 𝑥+1 𝑥−1 =0
𝑥 = −1,0,1
Make those critical points into
intervals:
−∞, −1 −1,0 0,1 (1, ∞)
Test point 𝑓(test point)
−2
−0.5
0.5
2
Is 𝒇(test point) positive or negative?
(−2)3 − −2 = −6
(−.5)3 − −.5 = .375
(.5)3 − .5 = −.375
(2)3 − 2 = 6
Negative
Positive
Negative
positive
Since our original equation asks when it our polynomial greater than 0, it is asking for
which intervals is 𝑓(test point) positive?
−1,0 and (1, ∞)
Solve the rational inequality:
𝑥(𝑥 − 3)
≥0
𝑥+2
interval
Test point
−∞, 0
0,3
(3, ∞)
−1
1
4
Critical Points:
𝑥 𝑥−3
𝑥+2
=0 𝑥+2
𝑥+2
𝑥 𝑥−3 =0
𝑥 = 0 𝐴𝑁𝐷 𝑥 − 3 = 0
𝑥 = 0,3
Make those critical points into
intervals: −∞, 0 0,3 (3, ∞)
𝑓(test point)
4
−.6666
. 6666
Is 𝒇(test point) positive or negative?
Positive
Negative
Positive
Since our original equation asks when it our polynomial greater than or equal to 0, it is
asking for which intervals is 𝑓(test point) positive or equal to zero?
(−∞, 0] and [3, ∞)
𝑥3 − 𝑥 > 0
(−1,0) ∪ (1, ∞)
𝑥(𝑥−3)
𝑥+2
≥0
−2,0 ∪ [3, ∞)
Direct and Inversely Proportional
Direct:
𝑦 is directly proportional to the 𝑛𝑡ℎ power of 𝑥
[𝑦 varies directly to as the 𝑛𝑡ℎ power of 𝑥] if
there exists a non-zero 𝑘 such that 𝑦 = 𝑘𝑥 𝑛
Inversely:
𝑦 is inversely proportional to the 𝑛𝑡ℎ power of 𝑥
[𝑦 varies inversely to as the 𝑛𝑡ℎ power of 𝑥] if
𝑘
there exists a non-zero 𝑘 such that 𝑦 = 𝑛
𝑥
Examples:
Find the constant of proportionality 𝑘:
𝑘
𝑦 = 𝑥 , and 𝑦 = 2 when 𝑥 = 3
𝑦 = 𝑘𝑥 3 and 𝑦 = 64 when 𝑥 = 2
Try these on your own!
6=𝑘
8=𝑘
Solve the variation problem: Try these on your own!
Suppose 𝑦 varies directly as the second power of 𝑥.
When 𝑥 = 3, 𝑦 = 10.8. Find 𝑦 when 𝑥 = 1.5.
𝑦 = 𝑘𝑥 2
1.2 = 𝑘
𝑦 = 2.7
Let 𝑦 be inversely proportional to 𝑥.
When 𝑥 = 6, 𝑦 = 5. Find 𝑦 when 𝑥 = 1.5.
𝑦=
𝑘
𝑥
30 = 𝑘
𝑦 = 45