GENMATH module 2-5 and 7 worksheet in general mathematics, Translations of Mathematics

Download GENMATH module 2-5 and 7 worksheet in general mathematics and more Translations Mathematics in PDF only on Docsity! Learning Module for General Mathematics 0 Republic of the Philippines Department of Education National Capital Region DIVISION OF CITY SCHOOLS – MANILA Manila Education Center Arroceros Forest Park Antonio J. Villegas St. Ermita, Manila GENERAL MATHEMATICS Quarter 1 Week 2 Module 4 Learning Competency: 1. Solves problems involving rational equations, and inequalities. M11GM-IC-3 1. Solves problems involving rational functions, equations and inequalities. M11GM-IC-3 1. Let’s think and act! Learning Module for General Mathematics 1 Before starting the module, I want you to set aside other tasks that will disturb you while enjoying the lessons. Read the simple instructions below to successfully enjoy the objectives of this kit. Have fun! 1. Follow carefully all the contents and instructions indicated in every page of this module. 2. Write on your notebook the concepts about the lessons. Writing enhances learning, that is important to develop and keep in mind. 3. Perform all the provided activities in the module. 4. Let your facilitator/guardian assess your answers using the answer key card. 5. Analyze conceptually the posttest and apply what you have learned. 6. Enjoy studying! HOW TO USE THIS MODULE? Learning Module for General Mathematics 4 B. Find the solution set for each inequality below. Graph the solution sets on the number line. 1. (𝑥𝑥+3)(𝑥𝑥−2) (𝑥𝑥+2)(𝑥𝑥−1) ≥ 0 2. (𝑥𝑥+4)(𝑥𝑥−3) (𝑥𝑥−2)(𝑥𝑥2+2) ≥ 0 3. 𝑥𝑥+1 𝑥𝑥+3 ≤ 2 4. 𝑥𝑥−2 𝑥𝑥2−3𝑥𝑥−10 < 0 Learning Module for General Mathematics 5 Steps in Solving Rational Equation 1. Find the value/s that will make the equation undefined. 2. Eliminate the denominators by multiplying each term by the lowest common denominator. 3. Solve the equation. 4. Check for any extraneous solution. Steps in Solving Rational Inequalities 1. Rewrite the inequality as a single fraction on one side of the inequality symbol and 0 on the other side. 2. Determine over what intervals the fraction takes on positive and negative values. a. Locate the x-values for which the rational expression is zero or undefined. Factor the numerator and denominator. b. Mark the numbers found in (a.) on a number line. Use a shaded circle to indicate that the value is included in the solution set, and a hollow circle to indicate that the value is excluded. These numbers divide the number line into intervals. c. Select a test point within the interior of each interval in (b.). the sign of the rational expression at this test point is also the sign of the rational expression at each interior point in the interval. d. Choose the interval that satisfy the sign of the inequality. Summarize the interval containing the solutions. Great, you finished answering the questions. You may request your facilitator to check your work. Congratulations and keep on learning! LOOKING BACK TO YOUR LESSON Learning Module for General Mathematics 6 Now we you already are a master in solving rational equations and inequalities! Indeed! You are now more than ready to solve problems involving these concepts. Usual problem that involves solving rational equation involves work-rate problems, water current-speed and more. Let’s now recall the process of solving problem word problems: STEPS IN SOLVING WORD PROBLEMS According to George Polya, known as the father of modern problem solving the following are steps to solve word problems: Step 1: Understand the problem Read the problem very carefully. List all the given data and identify variables that you can use in forming a mathematical equation to solve the problem. Step 2: Devise a plan (Translate) When you devise a plan, you are translating given data and variables into a Mathematical equation. Drawing a diagram or a chart can help you in analyzing given data to set up your equation. Step 3: Carry out the plan (Solve) The next step is to solve the equation. Step 4: Look back ( Check and Interpret) Last step is to check if the answer you obtained is correct. BRIEF INTRODUCTION Learning Module for General Mathematics 9 EXAMPLE 3: A box with a square base is to have a volume of 8 cubic meters. Let x be the length of the side of the square base and h be the height of the box. What are the possible measurements of a side of the square base if the height should be longer than a side of the square base? STEP 3: Set up the equation: Stated in the problem the phrase “It takes as long…” it means that the time he rowed is equal to each other, so we write 4 5−𝑐𝑐 = 16 5+𝑐𝑐 this can be easily solved by cross multiplying 4(5+c) =16(5-c) → 20+4c= 80-16c → 4c+16c =80-20 20c=60 → c=3 Thus, the speed of the current is 3 miles per hour. STEP 1: The volume of a rectangular box is the product of its width, length and height. The base of the box is square so the width and length are equal. Let x= length of the side of the box h= height of the box use the formula v = lwh to relate x and h v = 𝑥𝑥2ℎ since v= 8 → 8 = 𝑥𝑥2ℎ STEP 2: Express h in terms of x 8 = 𝑥𝑥2ℎ → h = 8 𝑥𝑥2 Learning Module for General Mathematics 10 STEP 3: Since the height is greater than the width, h> x and our inequality is 8 𝑥𝑥2 >x STEP 3: Since the height is greater than the width, h> x and our inequality is 8 𝑥𝑥2 >x STEP 4: Solve the inequality: 8 𝑥𝑥2 -x > 0 → rewrite the inequality the other side should be equal to 0 8−𝑥𝑥3 𝑥𝑥2 > 0 → express the rational expression as a single fraction (2−𝑥𝑥)(𝑥𝑥2+2𝑥𝑥+4) 𝑥𝑥2 > 0 → factor the numerator to find the critical values we get x=2 from the numerator and x = 0 from the denominator 0 2 Construct table of signs Interval x<0 0 < x < 2 x >2 Test Point x=-1 x=1 x=3 2 -x + + - 𝑥𝑥2 + 2𝑥𝑥 + 4 + + + 𝑥𝑥2 + + + (2 − 𝑥𝑥)(𝑥𝑥2 + 2𝑥𝑥 + 4) 𝑥𝑥2 + + - The solution set is 0 < x < 2, we reject the solution set in the interval x<0 even if the expression is positive, we will only consider positive integers because we are solving for the length of the box. Thus, the height of the box is less than 2 meters. Learning Module for General Mathematics 11 Activity 1. Using the given hint. Complete the following steps and solve the problem. PROBLEM: In a schools basketball league, the team from Recto High has won 12 out of 25 games, a winning percentage of 48%. How many games should they win in a row to improve their win percentage to 60% ACTIVITIES STEP 1: HINT: Let x =represent the number of games that they need to win to raise their percentage of winning to 60%. The team has already won 12 out of their 25 games. If they win x games in a row to increase their percentage to 60%, then they would have played 12+x games out of 25+x games the equation is Total win= (a)________ Total Number of Games = (b)___________ STEP 2: The equation is rational equation. Solve the equation. STEP 3: → HINT: to eliminate the denominator multiply the equation by (25+x) →HINT: distribute .6 to 25+x →HINT: Simplify →HINT: multiply the equation by 10 to get rid of the decimal → HINT: combine similar terms →HINT: divide both sides by 4 X= ________ Since x represents the number of games, this number should be an integer. Therefore Recto High needs to win _______ games in a row to raise their winning percentage to 60%. Learning Module for General Mathematics 14 Solve the following problems. Show every step for credit points. 1. John can build a pigpen twice as fast as Ruel. Working together, it takes them 5 hours. How long would it have taken John working alone? 2. A plane flies 910 miles with the wind in the same time it can go 660 miles against the wind. The speed of the plane in still air is 305 miles per hour. What is the speed of the wind? 3. A dressmaker ordered several meters of red cloth from a vendor, but the vendor only had 4 meters of red cloth in stock. The vendor bought the remaining lengths of red cloth from a wholesaler for P1120. He then sold those lengths of red cloth to the dressmaker along with the original 4 meters of cloth for a total of P1600. If the vendor’s price per meter is atleast P10 more than the wholesaler’s price per meter, what possible length of cloth did the vendor purchase from the wholesaler? POSTTEST Learning Module for General Mathematics 15 REFLECTIVE LEARNING SHEET ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ _____________________________________________________. Nothing is Impossible with God! In Mathematics, we have learned that in every problem there is a solution. In real life, we believe and we have faith that there is one God that always help us in our problems. In fact when we pray God listens to our prayers. As a senior high student, have you ever experience a moment in your life that God answered your kneeled down prayer? Share that answered prayer story. Write in 4 to 5 sentences. Learning Module for General Mathematics 16 To further explore the concept learned today and if it possible to connect the internet, you may visit the following links: https://www.youtube.com/watch?v=QLhvLEeS08A https://www.youtube.com/watch?v=puHW7GsmTuc https://www.youtube.com/watch?v=r6N8mDRNktw https://www.youtube.com/watch?v=IGGnn9oa4QY Versoza, et. Al, O. A. (2016). General Mathematics . CHED https://sccollege.edu/Faculty/epham/Documents/IntroAlgManuscript/Chapter%2011%20- %20rational%20equations%20and%20applications.pdf E-SITES REFERENCES