Embed Size (px)
Citation preview
1
Chapter 3:
Solving Rational
Equations
2
Day 1: Proportions
A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Warm-Up Evaluate each expression. Express the answer in simplest form.
1) 3
4×
5
6 2) 4 ×
5
12 3)
3
8+
1
4
A rational equation is an equation that contains two or more fractions. Rational equations differ from rational
expressions in that rational equations contain an equals sign (=) and must be solved, rather than simplified.
There are two types of rational equations: proportions, and equations that include sums and differences.
The Proportion Structure
⎕
⎕=
⎕
⎕
To solve: Cross-multiply.
Model Problem A
−𝑥
5= 4
Rewrite
for Structure:
Model Problem B
2
3𝑥 = 18
Rewrite:
Remember: Remember:
3
Model Problem C
Rewrite:
Remember:
Exercise Solve and check:
1. 𝑑
8= 6 2.
−𝑡
2= 12 3.
2𝑟
3= 16
4. −15 = −3𝑛
5 5.
2
5𝑏 + 6 = 10 6.
5
8𝑛 + 6 = 7
The first strategy to try when you see fractions is ___________________________________________
4
Binomials in Proportions
Model Problem A
𝑥 + 1
6=
4
5
Model Problem B
4(−1 − 𝑐)
8=
−2(𝑐 − 8)
−9
When multiplying a number by a binomial:
If the binomial has a coefficient already:
Model Problem C
2𝑥 − 1 =5
2
Model Problem D
15 = −1
2(−12𝑥 + 2)
Making a proportion: Dealing with parentheses on the side:
5
Quick Check for Understanding
1) 5
𝑥−3=
3
10 2)
3(𝑏−2)
4=
7
3
3) 9 − 2𝑑 =46
2 4)
7
6(6𝑝 − 24) + 9 = 23
Homework Chapter 2, Day 1 Complete in notebook.
1) 2
3𝑥 =
14
6
2) −4
5𝑔 − 3 = 9
3) 2𝑥−18
4=
3𝑥+1
2
4) 𝑥+9
5=
𝑥−7
10
5) 𝑥
3= 𝑥 + 4
6) 𝑦
6= 𝑦 + 5
7) 𝑡 − 6 =3
2𝑡
8) 𝑚 − 3 =4
5(𝑚 − 2)
9) 3
5(1 + 𝑝) =
21
20
10) 1
2(2ℎ − 1) =
1
3(2ℎ −
1
2)
6
Day 2: Equations with Sums or Differences
A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Warm-Up Solve for the value of b:
3𝑏 − 4 =8𝑏 − 11
2
Structure of Sums and Differences
⎕
⎕±
⎕
⎕=
⎕
⎕
To solve: Get a common denominator, then cross them out.
Solve the resulting equation. Check.
Model Problem A Numerical Denominators
𝑥
5 +
𝑥
2= 14
Guided Practice A
2𝑥
3 +
𝑥
6= 5
Check: Check: What I learned:
7
Model Problem B Variables on the Side
3
4𝑤 + 8 =
1
3𝑤 − 7
Check:
Strategies to Remember:
Guided Practice B
1 −5
8𝑥 = 2 −
2
3𝑥
Check:
What I learned:
8
Model Problem C Binomials in the Numerator
𝑡 − 3
6−
𝑡 − 25
5= 4
Check:
Strategies to Remember:
Guided Practice C
𝑎 − 4
2−
𝑎 − 6
4= 12
Check:
What I Learned:
9
Model Problem D Variables in Denominator
Check:
Strategies to Remember:
Guided Practice D
Check:
What I Learned:
10
Summary of Strategies
Structure of Sums and Differences: ⎕
⎕±
⎕
⎕=
⎕
⎕
Solving Method: Get a common denominator. Cross off denominators and solve.
o Variables/negatives/parentheses on side: Put them up top.
o Binomials in numerator: Put parentheses on them.
o Variables in denominator: Break the monomial down. Multiply by missing parts.
Homework Chapter 3, Day 2 Complete all problems in notebook.
1)
𝑥
3+
𝑥
7= 10
2) 𝑥
3+
𝑥
2= 5
3) 𝑥
2+
2
3𝑥 = 5
4) 3𝑦
4− 6 =
𝑦
8+ 4
5) 3𝑥
20+
1
10=
𝑥
4−
1
5
6) 𝑥−2
3+
2𝑥−4
4= 5
7) 𝑥+6
3−
𝑥−3
4= 2
8) 3𝑚+1
4= 2 −
3− 2𝑚
6
9) 2
𝑥+
4
3=
14
3𝑥
10) 1
𝑏−
1
4=
𝑏−4
2𝑏
Challenge!
11) 𝟏
𝒎−𝟑+
𝟐
𝒎−𝟑=
𝒎+𝟑
(𝒎−𝟑)(𝒎−𝟐)
11
Day 3: Practice Solving Rational Equations
A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Remember the Basic Structures:
A) Proportion B) Sums and Differences
⎕
⎕=
⎕
⎕
⎕
⎕±
⎕
⎕=
⎕
⎕
Match each equation with its basic structure. Write (A) or (B). Then solve each equation in the space provided.
1) 6
𝑡+3=
4
𝑡
2) 𝑎
4+
1
2=
2
3
3) 5
4(𝑟 − 3) = −
5
8
4) 𝑥 + 3
4=
12𝑥−7
12
Proportions Sums and Differences
12
Practice
1) 4
y=
1
y+6 2) −
4
5𝑡 +
2
5=
2
3
3) 𝑦
3− 8 = 1 4)
14−𝑐
6= 𝑐 − 7
13
5) 12
3(𝑎−2)=
8
𝑎+1 6)
1
4𝑝 +
2
5𝑝 =
1
2𝑝 −
9
20
7) 𝑥
2−
𝑥+3
8= 5 8)
5
𝑛2 −𝑛+4
𝑛2 =1
2𝑛2
14
9) −4
3(
5
3+ 𝑛) = −
16
9 10) 2𝑥 + 1 =
1
4(
1
2𝑥 + 4)
Challenge!
Homework Chapter 3, Day 3 Complete in your notebook.
1) 5
8=
𝑎
6
2) 𝑟
3−
𝑟
6= 2
3) −2
5𝑦 − 6 = 14
4) 3
4=
3
8𝑥 −
3
2
5) 2𝑥+1
3=
6𝑥−9
5
6) 𝑡−3
6−
𝑡−25
5= 4
7) 1
5(2𝑥 − 10) =
1
3(𝑥 − 1)
8) 𝑥−6
𝑥=
𝑥+4
𝑥+ 1
16
Day 4: More Practice Solving Rational Equations
A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Solve for the value of the variable.
1) 5𝑥
2=
15
4 2)
𝑥
3+
𝑥
5= 8
3) 1
3𝑥 + 8 = 18 4)
1
2𝑥 +
2
5𝑥 + 14 = 𝑥
17
5) 𝑚−2
9= 3 6) 𝑎 + 24 =
𝑎+8
4
7) 2𝑥+5
3−
𝑥−2
4= 24 8)
2
3(𝑥 + 5) = 6
18
9) 6𝑏+18
𝑏2+
1
𝑏=
3
𝑏 10)
1
3(−
7
4𝑘 + 1) −
10
3𝑘 =
13
8
Homework Chapter 3, Day 4
Part A: Textbook p. 96 #7-12 and p. 903 #2-9
Part B: These two problems –
1) 8
9(𝑥 + 6) = 𝑥 − 3
2) 2𝑥+7
6−
2𝑥−9
10= 3
19
Day 5: Review
A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Review of the Rules
Proportions
⎕⎕
=⎕⎕
Sums and Differences
⎕
⎕±
⎕
⎕=
⎕
⎕
If the structure matches,
just apply the rule. 1)
3𝑥
4=
12
20 2)
𝑎
2+
𝑎
3+
𝑎
4= 26
Always put binomials in
parentheses. 3) 5
9=
ℎ−26
ℎ+21 4)
𝑦+2
4−
𝑦−3
3=
1
2
20
Proportions
⎕⎕
=⎕⎕
Sums and Differences
⎕
⎕±
⎕
⎕=
⎕
⎕
Variables, negatives, and
parentheses go up top. 5)
3
2(𝑥 + 4) = 2(𝑥 − 2) 6) −
1
2𝑧 + 1 =
3
2
Level A
1) 2
5𝑟 = 8 2)
𝑥
3+
𝑥
4= 12 3)
𝑥−6
𝑥=
𝑥+4
𝑥+ 1
21
Level B
4) 3𝑦+1
2=
2(44−𝑦)
5 5)
1
2𝑥−
𝑥−1
2𝑥2 =3
𝑥 6)
1
6(2𝑥 + 7) −
1
10(2𝑥 − 9) = 3
Level C
7) 2𝑥 − 1 −2𝑥−2
2=
3𝑥+1
5+
𝑥+1
4 8) −
11
3+
3
2𝑏 =
5
2(𝑏 −
5
3)
