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 Your Turn