1. Adding/Subtracting Like Denominator Rational Expressions
Concept 9

2. Adding and Subtracting Rational Expressions with Common Denominators:
Adding and subtracting rational expressions is exactly like adding and subtracting fractions.
They must have common denominators.
Step 1: Make sure you have ______________ ________________.
Step 2: _______________ numerators and keep the ____________ ________________.
Step 3: _________ and ____________ the remaining expression!
common
denominators
ADD
common
denominators
Add
simplify

3. 𝑥 + 5 2𝑥
= 3+5 2𝑥
2. 𝑚−3𝑛 6 𝑚 3 𝑛 − 𝑚+3𝑛 6 𝑚 3 𝑛 4
= 𝑚−3𝑛−(𝑚+3𝑛) 6 𝑚 3 𝑛
= 8 2𝑥 x
= 𝑚−3𝑛−𝑚−3𝑛 6 𝑚 3 𝑛
= 4 𝑥 -1
= −6𝑛 6 𝑚 3 𝑛
= −𝑛 𝑚 3

4. 4. 2 𝑥 2 +2𝑥−15 𝑥−3 − 𝑥 2 +7𝑥+9 𝑥−3 3. 6 𝑥−1 − 𝑥+6 𝑥−1
𝑥−1 − 𝑥+6 𝑥−1
𝑥 2 +2𝑥−15 𝑥−3 − 𝑥 2 +7𝑥+9 𝑥−3
= 2 𝑥 2 +2𝑥−15−( 𝑥 2 +7𝑥+9) 𝑥−3
= 6 −(𝑥+6) 𝑥−1
= 2 𝑥 2 +2𝑥−15− 𝑥 2 −7𝑥−9 𝑥−3
= 6 −𝑥−6 𝑥−1
(𝑥−3)(𝑥+8)
= 𝑥 2 −5𝑥−24 𝑥−3
= −𝑥 𝑥−1 1
=𝑥+8

5. 𝑥−2 3𝑥 2 −19𝑥 𝑥 2 −19𝑥+6
𝑥 2 𝑥 2 −12𝑥+20 − 4𝑥 𝑥 2 −12𝑥+20
= 3𝑥−2+1 3𝑥 2 −19𝑥+6
2x(x−2)
= 2 𝑥 2 −4𝑥 𝑥 2 −12𝑥+20
= 3𝑥−1 3𝑥 2 −19𝑥+6 1
(x−10)(x−2)
𝑥 2 −19x+18
(x−18)(x−1)
= 2𝑥 𝑥−10
(x−6)(3x−1)
= 1 𝑥−6

6. 8. x x 2 −4 + 5x x 2 −4 − 2x+11 x 2 −4
𝑥 2 −𝑥 4𝑥 2 −25 − 15 4𝑥 2 −25
= x x−2x−11 x 2 −4
(x−3)(2x+5)
(x−6)(x+5)
𝑥 2 −x−30
= 2𝑥 2 −𝑥−15 4𝑥 2 −25
= x 2 +3x−10 x 2 −4
(x−2)(x+5)
(2x−5)(2x+5)
= 𝑥−3 2𝑥−5
(x−2)(x+2)
= 𝑥+5 𝑥+2

7. Adding and Subtracting Rational Expressions with Unlike Denominators:
Adding and subtracting rational expressions is exactly like adding and subtracting fractions.
They must have common denominators.
Step 1: ___________ everything before determining the ______________.
Step 2: Determine the ___________ ____________ and __________ the fractions.
Step 3: ___________ numerators and keep the ______________ ______ _________.
Step 4: _____________ any items.
Step 5: Combine _______ terms.
Step 6: ____________
Factor
denominator
common
denominator
rewrite
Combine
denominators
the
same
Add/Subtract
like
Simplify

8. 𝑦 6 − 𝑦 2
= 7𝑦 3∙2 − 𝑦 ∙3
𝑥 𝑥
= ∙𝑥 ∙5∙𝑥
3 * 2
4 * 5 * x
= 7𝑦 3∙2 − 3𝑦 ∙2
= ∙5∙𝑥 ∙5∙𝑥
= 7𝑦−3𝑦 3∙2
= ∙5∙𝑥 2y
= 4𝑦 3∙2 4
= 16 4∙5∙𝑥
= 2𝑦 3
= 4 5𝑥

9. 4. 12 4𝑘−20 − 1 𝑘−5 3. 𝑥−1 2 − 3𝑥+1 18 =9 𝑥−1 2 − 3𝑥+1 18 = 6𝑥−10 9∙2
𝑘−20 − 1 𝑘−5
9𝑥−9−3𝑥−1
9 𝑥−1 −(3𝑥+1)
3. 𝑥−1 2 − 3𝑥+1 18
=9 𝑥− − 3𝑥+1 18
4(𝑘−5)
= 6𝑥−10 9∙2
= 12−4 4(𝑘−5) 2
= 2(3𝑥 −5) 3∙2
= 8 4(𝑘−5)
= 3𝑥−5 9
= 2 𝑘−5

10. 6. 𝑦 𝑦−8 − 6𝑦+80 𝑦 2 −64 5. 𝑥−4 3𝑥−3 + 1 𝑥−1 = 𝑥−4+3 3(𝑥−1)
5. 𝑥−4 3𝑥−3 + 1 𝑥−1
6. 𝑦 𝑦−8 − 6𝑦+80 𝑦 2 −64
= 𝑥−4+3 3(𝑥−1)
(y – 8)(y + 8)
3(x – 1) 1
(y – 8)(y + 10)
= 𝑥−1 3(𝑥−1)
𝑦 2 +2𝑦−80
𝑦 2 +8𝑦−6𝑦−80
= 𝑦 𝑦+8 −(6𝑦+80) (𝑦−8)(𝑦+8)
= 1 3
= 𝑦+10 𝑦+8

11. 8. 3𝑥 𝑥 2 −4𝑥 − 1 𝑥−4 7. 6𝑥 𝑥 2 −4 − 3 𝑥−2 = 6𝑥 −3(𝑥+2) (𝑥−2)(𝑥+2)
𝑥 𝑥 2 −4 − 3 𝑥−2
8. 3𝑥 𝑥 2 −4𝑥 − 1 𝑥−4
= 6𝑥 −3(𝑥+2) (𝑥−2)(𝑥+2)
𝑥(𝑥−4)
(𝑥−2)(𝑥+2)
= 6𝑥 −3𝑥−6 (𝑥−2)(𝑥+2)
= 3𝑥−𝑥 𝑥(𝑥−4)
3(x – 2)
= 2𝑥 𝑥(𝑥−4)
= 3𝑥 −6 (𝑥−2)(𝑥+2)
= 2 𝑥−4
= 3 𝑥+2

12. 9. 3 𝑥+1 − 6 𝑥 2 +4𝑥+3 = 3 𝑥+3 −6 𝑥+3 (𝑥+1) = 3𝑥+9−6 𝑥+3 (𝑥+1)
𝑥+1 − 6 𝑥 2 +4𝑥+3
= 3 𝑥+3 −6 𝑥+3 (𝑥+1)
= 3𝑥+9−6 𝑥+3 (𝑥+1)
(x + 3)(x + 1)
3(x + 1)
= 3𝑥+3 𝑥+3 (𝑥+1)
= 3 𝑥+3
2(x – 6)
𝑥−6 + 𝑥−17 𝑥 2 −𝑥−30
= 𝑥+5+𝑥−17 (𝑥+5)(𝑥−6)
= 2𝑥−12 (𝑥+5)(𝑥−6)
(x + 5)(x – 6)
= 2 𝑥+5

13. 11. 8𝑥−26 𝑥 2 −4𝑥−21 − 3 𝑥−7 = 8𝑥−26 −3(𝑥+3) (𝑥−7)(𝑥+3)
𝑥−26 𝑥 2 −4𝑥−21 − 3 𝑥−7
= 8𝑥−26 −3(𝑥+3) (𝑥−7)(𝑥+3)
= 8𝑥−26 −3𝑥−9 (𝑥−7)(𝑥+3)
(x – 7)(x + 3)
7(x – 7)
= 5𝑥−35 (𝑥−7)(𝑥+3)
= 7 𝑥+3
𝑥 2 −3𝑥−1 2𝑥 2 +5𝑥+2 + 𝑥 2𝑥+1
= 𝑥 2 −3𝑥−1+𝑥(𝑥+2) (𝑥+2)(2𝑥+1)
(𝑥−1)(2𝑥+1)
𝑥 2 +5𝑥+4
= 𝑥 2 −3𝑥−1+ 𝑥 2 +2𝑥 (𝑥+2)(2𝑥−1)
= 2 𝑥 2 −𝑥−1 (𝑥−2)(2𝑥−1)
(𝑥+4)(𝑥+1)
(𝑥+2)(2𝑥+1)
= 𝑥−1 𝑥−2

14. Solving rational Equations
Concept 11

15. 𝟏. 𝟒 𝒙+𝟏 = 𝟑 𝒙 𝟑. 𝟑 𝟐𝒙−𝟏 = 𝟐 𝒙−𝟒 4 𝑥 =3(𝑥+1) 3(𝑥 −4)=2(2x −1) 4𝑥=3𝑥+3
𝟏. 𝟒 𝒙+𝟏 = 𝟑 𝒙
𝟑. 𝟑 𝟐𝒙−𝟏 = 𝟐 𝒙−𝟒
4 𝑥 =3(𝑥+1)
3(𝑥 −4)=2(2x −1)
4𝑥=3𝑥+3
3𝑥 −12=4x −2
𝑥=3
−10=x
𝟐 𝒙 + 𝟑 𝟐
= 𝟐 𝟐 +𝟑𝒙 𝟐𝒙
𝟐. 𝟓 𝒚−𝟏 = 𝟐 𝒚
𝟒. 𝟐 𝒙 + 𝟑 𝟐 = 𝟓 𝟐
= 𝟒+𝟑𝒙 𝟐𝒙 2x
𝟒+𝟑𝒙 𝟐𝒙 = 𝟓 𝟐
5 𝑦 =2(𝑦−1)
5𝑦=2𝑦−2
8=4x
2 4+3𝑥 =5(2x)
3𝑦=−2
2=x
𝑦= −2 3
8+6𝑥=10x

16. 𝑥 = 3 𝑥
𝑥 = 4 3𝑥
𝟐 𝟑 + 𝟏 𝒙
= 𝟐 𝒙 +𝟑 𝟑𝒙
𝟏 𝟑𝒙 + 𝟏 𝟐
= 𝟏 𝟐 +𝟏(𝟑𝒙) 𝟔𝒙
= 𝟐𝒙+𝟑 𝟑𝒙 3x
= 𝟐𝒙+𝟑 𝟔𝒙 6x
𝟐𝒙+𝟑 𝟑𝒙 = 𝟑 𝒙
2𝑥+3 6𝑥 = 4 3𝑥 1
2𝑥+3=3(3) 2 1
2𝑥+3=9
2𝑥+3=4(2)
2𝑥=6
2𝑥+3=8
𝑥=3
2𝑥=5
𝑥= 5 2 =2.5

17. 𝟕. 𝟖+ 𝟐 𝒙−𝟏 = 𝟐 𝒙−𝟏 𝟖. 𝟏 𝒙−𝟏 +𝟏= 𝒙 𝒙+𝟏 𝟖𝒙 −𝟔 𝒙−𝟏 = 𝟐 𝒙−𝟏 𝒙 𝒙−𝟏 = 𝒙 𝒙+𝟏
𝟕. 𝟖+ 𝟐 𝒙−𝟏 = 𝟐 𝒙−𝟏
𝟖. 𝟏 𝒙−𝟏 +𝟏= 𝒙 𝒙+𝟏
= 𝟖 𝒙−𝟏 +𝟐 𝒙−𝟏
= 𝟏+(𝒙−𝟏) 𝒙−𝟏
𝟖 𝟏 + 𝟐 𝒙−𝟏
= 𝟖𝒙 −𝟖+𝟐 𝒙−𝟏
= 𝒙 𝒙−𝟏
x - 1 1 1
= 𝟖𝒙 −𝟔 𝒙−𝟏
𝟖𝒙 −𝟔 𝒙−𝟏 = 𝟐 𝒙−𝟏
𝒙 𝒙−𝟏 = 𝒙 𝒙+𝟏 1 1
𝑥−1=x+1
−1=1
𝑛𝑜 𝑟𝑒𝑎𝑙 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠
8𝑥−6=2
8𝑥=8
𝑥=1

18. 𝟗. 𝟐 𝒙−𝟏 + 𝟑 𝒙 =𝟐 𝟏𝟎.𝟏− 𝟐 𝒙+𝟐 = 𝟐 𝒙−𝟐 𝒙 𝒙+𝟐 = 𝟐 𝒙−𝟐 𝒙 𝒙−𝟐 =𝟐(𝒙+𝟐)
𝟗. 𝟐 𝒙−𝟏 + 𝟑 𝒙 =𝟐
𝟏𝟎.𝟏− 𝟐 𝒙+𝟐 = 𝟐 𝒙−𝟐
= 𝟏 𝒙+𝟐 −𝟐 𝒙+𝟐
= 𝟐(𝒙)+𝟑(𝒙−𝟏) 𝒙(𝒙−𝟏)
𝒙 𝒙+𝟐 = 𝟐 𝒙−𝟐
𝒙 𝒙−𝟐 =𝟐(𝒙+𝟐)
𝒙 𝟐 −𝟐𝒙=𝟐𝒙+𝟒
𝒙 𝟐 −𝟒𝒙−𝟒=𝟎
= 𝒙 𝒙+𝟐
= 𝟐𝒙+𝟑𝒙−𝟑 𝒙(𝒙−𝟏)
𝟓𝒙−𝟑 𝒙(𝒙−𝟏) = 𝟐 𝟏
0=2𝑥−1
1=2𝑥
1 2 =𝑥
5𝑥−3=2𝑥(𝑥−1)
5𝑥−3=2 𝑥 2 −2𝑥
0=2 𝑥 2 −7𝑥+3
0= 𝑥 2 −7𝑥+6
0=(𝑥−1)(𝑥−6)
0=(2𝑥−1)(𝑥−3)
−(−𝟒)± (−𝟒) 𝟐 −𝟐(𝟏)(−𝟒) 𝟐(𝟏)
𝟒± 𝟏𝟔+𝟖 𝟐
= 𝟒± 𝟐𝟒 𝟐
0=𝑥−3
3=𝑥

19. 𝟏𝟏. 𝟑 𝒙 𝟐 +𝟐𝒙+𝟏 + 𝟐 𝒙+𝟏 =𝟏 𝟏𝟐. 𝒙−𝟏 𝒙+𝟑 + 𝒙 𝒙−𝟒 = −𝟒 𝒙+𝟑
𝟏𝟏. 𝟑 𝒙 𝟐 +𝟐𝒙+𝟏 + 𝟐 𝒙+𝟏 =𝟏
𝟏𝟐. 𝒙−𝟏 𝒙+𝟑 + 𝒙 𝒙−𝟒 = −𝟒 𝒙+𝟑
(x + 1)(x + 1)
= 𝒙−𝟏 (𝒙−𝟒)+𝒙(𝒙+𝟑) (𝒙+𝟑)(𝒙−𝟒)
= 𝟑+𝟐(𝒙+𝟏) (𝒙+𝟏)(𝒙+𝟏)
= 𝒙 𝟐 −𝟓𝒙+𝟒+ 𝒙 𝟐 +𝟑𝒙 (𝒙+𝟑)(𝒙−𝟒)
= 𝟑+𝟐𝒙+𝟐 (𝒙+𝟏)(𝒙+𝟏)
𝟐𝒙 𝟐 −𝟐𝒙+𝟒 (𝒙+𝟑)(𝒙−𝟒) = −𝟒 𝒙+𝟑
2𝑥+5 𝑥+1 (𝑥+1) = 1 1
𝟐 𝒙 𝟐 −𝟐𝒙+𝟒=−𝟒(𝐱−𝟒)
𝟐 𝒙 𝟐 −𝟐𝒙+𝟒=−𝟒𝐱+𝟏𝟔
𝟐𝒙 𝟐 +𝟐𝒙−𝟏𝟐=𝟎
𝟐𝒙+𝟓= 𝒙 𝟐 +𝟐𝒙+𝟏
𝟒= 𝒙 𝟐
2 = x

20. 13. 𝑥−1 𝑥+5 + 𝑥+3 𝑥+4 = 𝑥+1 𝑥+4