Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
Lesson Summary
To compare two rational expressions, find equivalent rational expression with the same denominator. Then we can
compare the numerators for values of the variable that do not cause the rational expression to change from
positive to negative or vice versa.
We may also use numerical and graphical analysis to help understand the relative sizes of expressions.
Problem Set
1.
For parts (a)–(d), rewrite each rational expression as an equivalent rational expression so that all expressions have a
common denominator.
a.
b.
c.
d.
3 9
,
,
7
,
7
5 10 15 21
𝑚𝑚 𝑠𝑠 𝑑𝑑
,
,
𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑 𝑚𝑚𝑚𝑚
1
,
3
(2−𝑥𝑥)2 (2𝑥𝑥−5)(𝑥𝑥−2)
3
5 2𝑥𝑥+2
, ,
𝑥𝑥−𝑥𝑥 2 𝑥𝑥 2𝑥𝑥 2 −2
1
𝑥𝑥
2.
If 𝑥𝑥 is a positive number, for which values of 𝑥𝑥 is 𝑥𝑥 < ?
3.
Can we determine if
4.
For positive 𝑥𝑥, determine when the following rational expressions have negative denominators.
a.
b.
c.
d.
3
5
𝑦𝑦
𝑦𝑦−1
>
𝑦𝑦+1
𝑦𝑦
for all values 𝑦𝑦 > 1? Provide evidence to support your answer.
𝑥𝑥
5−2𝑥𝑥
𝑥𝑥+3
𝑥𝑥 2 +4𝑥𝑥+8
3𝑥𝑥 2
(𝑥𝑥−5)(𝑥𝑥+3)(2𝑥𝑥+3)
Lesson 23:
© 2015 Great Minds.. eureka-math.org
ALG II-M1-SE-1.3.0-06.2015
Comparing Rational Expressions
S.117
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
5.
Consider the rational expressions
a.
b.
𝑥𝑥
𝑥𝑥−2
and
Evaluate each expression for 𝑥𝑥 = 6.
𝑥𝑥
.
𝑥𝑥−4
c.
Evaluate each expression for 𝑥𝑥 = 3.
d.
Extension: Raphael claims that the calculation below shows that
Can you conclude that
𝑥𝑥
𝑥𝑥−2
<
𝑥𝑥
𝑥𝑥−4
for all positive values of 𝑥𝑥? Explain how you know.
and 𝑥𝑥 ≠ 4. Where is the error in the calculation?
Starting with the rational expressions
𝑥𝑥
𝑥𝑥−2
and
𝑥𝑥
𝑥𝑥
𝑥𝑥−2
<
𝑥𝑥
𝑥𝑥−4
for all values of 𝑥𝑥, where 𝑥𝑥 ≠ 2
, we need to first find equivalent rational expressions with a
𝑥𝑥−4
common denominator. The common denominator we will use is (𝑥𝑥 − 4)(𝑥𝑥 − 2). We then have
𝑥𝑥(𝑥𝑥 − 4)
𝑥𝑥
=
𝑥𝑥 − 2 (𝑥𝑥 − 4)(𝑥𝑥 − 2)
𝑥𝑥(𝑥𝑥 − 2)
𝑥𝑥
=
.
𝑥𝑥 − 4 (𝑥𝑥 − 4)(𝑥𝑥 − 2)
Since 𝑥𝑥 2 − 4𝑥𝑥 < 𝑥𝑥 2 − 2𝑥𝑥 for 𝑥𝑥 > 0, we can divide each expression by (𝑥𝑥 − 4)(𝑥𝑥 − 2). We then have
𝑥𝑥(𝑥𝑥−4)
(𝑥𝑥−4)(𝑥𝑥−2)
6.
<
𝑥𝑥(𝑥𝑥−2)
(𝑥𝑥−4)(𝑥𝑥−2)
, and we can conclude that
𝑥𝑥
𝑥𝑥−2
<
𝑥𝑥
𝑥𝑥−4
for all positive values of 𝑥𝑥.
Consider the populations of two cities within the same state where the large city’s population is 𝑃𝑃, and the small
city’s population is 𝑄𝑄. For each of the following pairs, state which of the expressions has a larger value. Explain your
reasoning in the context of the populations.
a.
b.
c.
d.
e.
f.
g.
h.
𝑃𝑃 + 𝑄𝑄 and 𝑃𝑃
𝑃𝑃
𝑃𝑃+𝑄𝑄
and
𝑄𝑄
𝑃𝑃+𝑄𝑄
2𝑄𝑄 and 𝑃𝑃 + 𝑄𝑄
𝑃𝑃
𝑄𝑄
and
𝑃𝑃
𝑃𝑃+𝑄𝑄
𝑃𝑃+𝑄𝑄
𝑃𝑃
𝑃𝑃+𝑄𝑄
1
𝑃𝑃
2
𝑄𝑄
𝑃𝑃
and
1
2
and 𝑃𝑃 − 𝑄𝑄
and
and
1
𝑄𝑄
𝑃𝑃+𝑄𝑄
𝑄𝑄
Lesson 23:
© 2015 Great Minds.. eureka-math.org
ALG II-M1-SE-1.3.0-06.2015
Comparing Rational Expressions
S.118
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.