Adding/Subtracting Like Denominator Rational Expressions Concept 9

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

1. 3 2𝑥 + 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. 2 𝑥 2 +2𝑥−15 𝑥−3 − 𝑥 2 +7𝑥+9 𝑥−3 3. 6 𝑥−1 − 𝑥+6 𝑥−1 3. 6 𝑥−1 − 𝑥+6 𝑥−1 4. 2 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. 3𝑥−2 3𝑥 2 −19𝑥+6 + 1 3𝑥 2 −19𝑥+6 6. 2 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

8. x 2 +1 x 2 −4 + 5x x 2 −4 − 2x+11 x 2 −4 7. 2 𝑥 2 −𝑥 4𝑥 2 −25 − 15 4𝑥 2 −25 = x 2 +1+5x−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

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

1. 7𝑦 6 − 𝑦 2 = 7𝑦 3∙2 − 𝑦 2 ∙3 2. 3 4𝑥 + 1 20𝑥 = 5 3 4∙𝑥 + 1 4∙5∙𝑥 3 * 2 4 * 5 * x = 7𝑦 3∙2 − 3𝑦 3∙2 = 15 4∙5∙𝑥 + 1 4∙5∙𝑥 = 7𝑦−3𝑦 3∙2 = 15+1 4∙5∙𝑥 2y = 4𝑦 3∙2 4 = 16 4∙5∙𝑥 = 2𝑦 3 = 4 5𝑥

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

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

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

9. 3 𝑥+1 − 6 𝑥 2 +4𝑥+3 = 3 𝑥+3 −6 𝑥+3 (𝑥+1) = 3𝑥+9−6 𝑥+3 (𝑥+1) 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 + 3)(x + 1) 3(x + 1) = 3𝑥+3 𝑥+3 (𝑥+1) = 3 𝑥+3 2(x – 6) 10. 1 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

11. 8𝑥−26 𝑥 2 −4𝑥−21 − 3 𝑥−7 = 8𝑥−26 −3(𝑥+3) (𝑥−7)(𝑥+3) 11. 8𝑥−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 12. 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

Solving rational Equations Concept 11

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

5. 2 3 + 1 𝑥 = 3 𝑥 6. 1 3𝑥 + 1 2 = 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

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

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

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

13. 𝑥−1 𝑥+5 + 𝑥+3 𝑥+4 = 𝑥+1 𝑥+4