LP Math
Solving Equations
Containing Rational
Expressions
Solving Equations Containing Rational
Expressions Example 1
Justificatio
n
Step-by-step
solution
Solve the equation for N.
(Provided direction)
(Solve literal equations much
9
P = N + 25
the same way to isolate or solve
7
9
for a standard variable by
P − 25 = N
7
undoing what is happening to
𝟕
9 the 𝟕
∙ (P − 25) = N ∙ variable. In this case the
𝟗
7 addition
𝟗 of 25 and
7P − 175
7P multiplication
175
of 9/7)
N=
𝑜𝑟
−
9
9
9
Answer
(Simplify)
Solving Equations Containing Rational
Expressions Example 2
Justification
Step-by-step solution
Solve the equation for the
specified variable.
A−C
I=
for C
L
A−C
L∙I=
∙L
L
(Provided direction)
(Solve literal equations much
the same way to isolate or
solve for a standard variable
by undoing what is happening
IL − 𝐀 = A − C − 𝐀
to the variable.)
−C = IL − A which is C = A − IL
Answer
(Simplify)
Solving Equations Containing Rational
Expressions Example 3
Step-by-step solution
Justification
Solve the equation for 𝑓. (Provided direction)
(Solve literal equations much
𝑓(a + d)
S=
2
the same way to isolate or solve
𝑓(a + d) for a standard variable by
2∙S=
∙2
2
undoing what is happening to
2S
𝑓(𝑎 + 𝑑)
=
the variable.)
(𝑎 + 𝑑)
(𝑎 + 𝑑)
2S
𝑓=
(𝑎 + 𝑑)
Answer
(Simplify; Note all literal
equations are case sensitive)
Solving Equations Containing Rational
Expressions Example 4
Justification
Step-by-step solution
Solve the equation for 𝑠.
(Provided direction)
(Solve literal equations much
𝑓(s + z)
A=
the same way to isolate or solve
11
𝑓(s + z)
for a standard variable by
11 ∙ A =
∙ 11
11
undoing what is happening to
the+variable.)
11𝐴 = 𝑓(𝑠 + z) → 11𝐴 = 𝑓𝑠
𝑓z
11𝐴 − 𝑓𝑧 = 𝑓𝑠 →
(Simplify;
11𝐴
− 𝑓𝑧 Note: can you think of
𝑠=
another
way to represent this
𝑓
Answer
answer?)
Solving Equations Containing Rational
Expressions Example 5
Step-by-step solution
Solve the equation for 𝑠.
as
N=
a+s
Justification
(Provided direction)
(Solve literal equations much
the=same
N a + s = as → Na + Ns
as way to isolate or
solve for a standard variable
Na = as − Ns Na = s(a −byN)undoing what is happening
Na
s=
a−N
Answer
to the variable.)
(Simplify)
Solving Equations Containing Rational
Expressions Example 6
Step-by-step solution
A camel can drink 15 gallons of
water in 10 minutes. At this rate,
how much water can the camel
drink in 2 minutes?
15g
𝑔
=
10mi𝑛 2𝑚𝑖𝑛
10g = 30
g=3
Answer
Justification
(Provided direction)
(Set up for these scena
proportions in that
(Simplify)
𝑔𝑎𝑙
𝑚𝑖𝑛
Solving Equations Containing Rational
Expressions Example 7
Step-by-step solution
Justificatio
n
In 2005, 13.7 out of every 50
employees at a company were
women. If there are 44,191 total
company employees, estimate the
number of women.
(Provided direction)
(Set up for these scena
proportions in that
13.7
𝑤
=
50
44,191
50w = 605,416.7
w = 12,108.334
Answer
w = 12,108
(Simplify)
𝑝𝑎
𝑤ℎ𝑜
Solving Equations Containing Rational
Expressions Example 8
Step-by-step solution
Justification
A giant tortoise can travel
0.14 miles in 1 hour. At
this rate, how long would it
take the tortoise to travel 3
miles ?
0.14mi 3mi
=
1hr
h
0.14h = 3
h = 21.4
Answer
(Provided direction)
(Set up for these scena
proportions in that
(Simplify)
𝑝𝑎
𝑤ℎ𝑜
Solving Equations Containing Rational
Expressions Example 9
Step-by-step solution
Solve the following equation.
(𝑥 − 1)(𝑥 − 2)
Justification
(Provided direction
5
3
−1
(𝑥 − 1)(𝑥 − 2) (Multiply both sides
−
= 2
𝑥 − 1 𝑥 − 2 𝑥 − 3𝑥 + 2
is 𝑥 2 − 3𝑥 + 2 = (𝑥
5 𝑥 − 2 − 3 𝑥 − 1 = −1
(Simplify)
5𝑥 − 10 − 3𝑥 + 1 = −1
2𝑥 − 9 = −1
2𝑥 = 8 → 𝑥 = 4
Answer
Solving Equations Containing Rational
Expressions Example 10
Step-by-step solution
Solve the following equation.
Justification
(Provided direction
4
5𝑥
5
(𝑥 + 12)(𝑥 − 12) (Multiply both sides
− 2
=
𝑥 − 12 𝑥 − 144 𝑥 + 12
is 𝑥 2 − 144 = (𝑥 +
4 𝑥 + 12 − 5𝑥 = 5(𝑥 − 12)
(𝑥 + 12)(𝑥 − 12)
4𝑥 + 48 − 5𝑥 = 5𝑥 − 60
−𝑥 + 48 = 5𝑥 − 60
6𝑥 = 108 → 𝑥 = 18
Answer
(Simplify)
Solving Equations Containing Rational
Expressions Example 11
Step-by-step solution
Solve the following equation.
Justification
(Provided direction
1
4
− 2
𝑥(𝑥 − 4) ∙ (
)= 1 ∙ 𝑥(𝑥 − 4) (Multiply both sides
𝑥 − 4 𝑥 − 4𝑥
is 𝑥(𝑥 − 4))
𝑥 − 4 = 𝑥 2 − 4𝑥
𝑥 2 − 5𝑥 + 4 = 0
(Simplify and since
(𝑥 − 4)(𝑥 − 1) = 0
because that would
𝑥 − 1 = 0 → 𝑥 = 1, 𝑥 − 4 = 0 → 𝑥 = 4 denominator, we dis
for being extraneou
𝑥=1
Answer
Solving Equations Containing Rational
Expressions Example 12
Step-by-step solution
Justification
Solve the following equation.
𝑥2 − 3
3(𝑥 − 1)
𝑥∙(
+ 4)=
∙𝑥
𝑥
𝑥
(Provided direction
(Multiply both sides
𝑥 2 − 3 + 4𝑥 = 3(𝑥 − 1)
is 𝑥)
𝑥 2 − 3 + 4𝑥 = 3𝑥 − 3
(Simplify and since
𝑥2 + 𝑥 = 0
because that would
𝑥 𝑥 + 1 = 0 → 𝑥 = 0, −1
𝑥 = −1
Answer
denominator we dis
for being extraneou
Solving Equations Containing Rational
Expressions Example 13
Step-by-step solution
Justification
Solve the following equation.
−8
(3𝑦 + 1) ∙ (
+ 3)= 𝑦 ∙ (3𝑦 + 1)
3𝑦 + 1
−8 + 9𝑦 + 3 = 3𝑦 2 + 𝑦
3𝑦 2 − 8𝑦 + 5 = 0
(3𝑦 − 5)(𝑦 − 1) = 0
5
3𝑦 − 5 = 0 → 𝑦 = ; 𝑦 − 1 = 0 → 𝑦 = 1
3
5
𝑦 = ,1
3
Answer
(Provided direction
(Multiply both sides
is 3𝑦 + 1)
(Factor and solve)
Solving Equations Containing Rational
Expressions Example 14
Step-by-step solution
Justification
Solve the following equation.
(𝑥 + 3)(𝑥 − 3)
(Provided direction
44
2𝑥
8
+
+
=0
2
𝑥 −9 𝑥−3 𝑥+3
44 + 2𝑥 𝑥 + 3 + 8 𝑥 − 3 = 0
(𝑥 + 3)(𝑥 − 3)
(Multiply both sides
is 𝑥 2 − 9 = (𝑥 + 3)(
(Simplify)
44 + 2𝑥 2 + 6𝑥 + 8𝑥 − 24 = 0
2𝑥 2 + 14𝑥 + 20 = 0
(Factor)
2
2(𝑥 + 7𝑥 + 10) = 0
2(𝑥 + 5)(𝑥 + 2) = 0
(Simplify)
𝑥 + 5 = 0 → 𝑥 = −5 𝑥 + 2 = 0 → 𝑥 = −2
Answer
Answer
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