Solving Rational Equations & Inequalities Section 5.5

Rational Equations Equation that contains 1 or more rational expression To solve: Multiply each term by LCD All fractions should be gone now Solve the equation using inverse operations

EXAMPLES EX: Solve the equation. Check your solution. 1) ) 6 𝑥 + 5 4 =− 7 4 2) 𝑥− 18 𝑥 =3 3) 5𝑥 𝑥−2 = 3𝑥+4 𝑥−2 4) 16 𝑥 2 −16 = 2 𝑥−4 5) 10 3 = 4 𝑥 +2 6) 2𝑥−5 𝑥−8 + 𝑥 2 = 11 𝑥−8

Word Problems You need to determine a formula that relates the information in the given problem 𝑑=𝑟𝑡 Person A’s rate times combined hours + Person B’s rate times combined hours = 1 job Rates will always be 1 over hours can complete job independently

EXAMPLES 1) A jet travels 3950 miles from Chicago, Illinois, to London, England, and 3950 miles on the return trip. The total flying time is 16.5 hours. The return trip takes longer due to winds that generally blow from west to east. If the jet’s average speed with no wind is 485 mi/h, what is the average speed of the wind during the round-trip flight? Round to the nearest mile per hour 2) Natalie can finish a 500-piece puzzle in about 8 hours. When Natalie and Renzo work together, they can finish a 500-piece puzzle in about 4.5 hours. About how long will it take Renzo to finish a 500-piece puzzle if he works by himself?

Rational Inequalities To solve Algebraically Must consider 2 cases LCD positive Original inequality and LCD>0 LCD negative Opposite inequality symbol and LCD<0 Solve original inequality and LCD>0 Where are both of these solutions satisfied? Flip both solution’s inequality symbols

EXAMPLES 4 𝑥+1 <4 LCD: 𝑥+1 EX: Solve the inequality algebraically. Solving inequality gets us 4<4𝑥+4 which is 𝑥>0. Solving LCD > 0 gets us 𝑥>−1 𝑥>0 and 𝑥>−1 when 𝑥>0 Flipped solutions: 𝑥<0 and 𝑥<−1 when 𝑥<−1 Final answer: 𝑥<−1 OR 𝑥>0 EX: Solve the inequality algebraically. 1) 6 𝑥−8 ≤3 2) 6 𝑥−2 >−4

Rational Inequalities To Solve using a graph or table: Enter the left side of the inequality into Y1 any quantities in numerator or denominator need to go in ( ) Enter the right side of the inequality into Y2 Use graph or table to compare both sides of inequality to each other based on the inequality symbol in the problem. Table: Looking for ERROR and when Y1 = Y2 Graph: 2nd trace to calculate intersection and need to identify x-value where fraction is undefined Your solution needs to be written as an inequality EX: Use a graph or table to solve. 1) 𝑥 𝑥−6 ≤3 2) 𝑥 𝑥−3 ≥4

HOMEWORK Pages 353-355 #20-28even, 29, 30, 32, 33-37, 38-46even, 47-49, 54-56 Accelerated add #58-61