2_Q1-General-Mathematics.pptx qfor grade

math for g 11 perfect for students

1. General mathematics Rational Functions

2. Test I: Choose the letter of the correct answer and write it on a separate sheet of paper. 1. Martin can finish a job in 6 hours working alone. Victoria has more experience and can finish the same job in 4 hours working alone. How long will it take both people to finish that job working together? A. 2.4 hours B. 2.9 hours C. 3.5 hours D. 3.7 hours 2. Sarah can finish a job in 7 hours working alone. If Sarah and Matteo work together, they can finish the work in 3 hours. How long will it take if Matteo will choose to work alone? A. 4.25 hours B. 4.85 hours C. 5.25 hours D. 5.55 hours

3. Real Life Functions LESSON 5

4. How do we find the Least Common Denominator (LCD)? To add or subtract fractions with different denominators, you must find the least common denominator. LCD refers to the lowest multiple shared by each original denominator in the equation, or the smallest whole number that can be divided by each denominator.

5. How to solve problems involving Rational Functions? Example 1: Martin can finish a job in 6 hours working alone. Victor has more experience and can finish the same job in 4 hours working alone. How long will it take both people to finish that job working together?

6. Given: 6 hours – Martin can do the work alone 4 hours – Victor can do the work alone Find: x – hours Martin and Victor can do the work Solution:

7. Therefore, it will take 2.4 hours for Martin and Victor to do the work together.

8. Example 2: Sarah can finish a job in 7 hours working alone. If Sarah and Matteo work together, they can finish the work in 3 hours. How long will it take if Matteo will do the work alone? Given: 7 hours – Sarah can do the work alone 3 hours – Sarah and Matteo can do the work together Find: x hours – for Matteo to do the work alone

9. Solution:

10. NOW IT’S YOUR TURN! Directions: Solve each problem. Show your solutions and write answers on a separate sheet of paper. One person can complete a task in 8 hours. Another person can complete a task in 3 hours. How many hours does it take for them to complete the task if they work together? 2. Sigfried can paint a house in 5 hours. Stephanie can do it in 4 hours. How long will it take the two working together?

11. 3. Joy can pile 100 boxes of goods in 5 hours. Stephen and Joy can pile 100 boxes in 2 hours. If Stephen chooses to work alone, how long will it take? 4. Computer A can finish a calculation in 20 minutes. If Computer A and Computer B can finish the calculation in 8 minutes, how long does it take for the Computer B to finish the calculation alone?

12. Remember…

13. Rational Functions, Equations and Inequalities Lesson 6

14. The table below shows the definitions of rational functions, rational equations and rational inequalities with examples. DEFINITION OF TERMS

15. NOW IT’S YOUR TURN! Directions: Identify whether the following is a rational function, rational equation or rational inequality.

16. • Rational Function is a function of the form of 𝑓(𝑥)=𝑝(𝑥)𝑞(𝑥) where p(x) and q(x) are polynomials, and q(x) is not the zero function. • Rational Equation is an equation involving rational expressions. • Rational Inequality is an inequality involving rational expressions. Remember…

17. Solving Rational Equations and Inequalities LESSON 7

18. Factoring Example 1: Factor 2𝑥+6 → common factor is 2. 2 (x + 3) Therefore, the factors are 2 and x + 3 REVIEW

19. Example 2: Factor 3𝑥2+12𝑥 → common factor is 3x. 3x (x + 4) Therefore, the factors are 3x and x + 4

20. Example 3: Factor 𝑥2+8𝑥+15 Since there is no common factor, use factoring trinomials So, we can factor the whole expression into x2 + 8x + 15 = (x + 3)(x + 5)

21. Example 4: Factor 𝑥2−16 Since there is no common factor, use the factoring of sum and difference of two squares So, we can factor the whole expression into x2 - 16 = (x + 4)(x - 4)

22. What is the difference between Rational Equation and Inequalities?

23. A rational equation is an equation that contains one or more rational expressions while a rational inequality is an inequality that contains one or more rational expressions with inequality symbols ≤, ≥, <, >, and ≠.

24. Solving Rational Equations Example 1: Solve for 𝑥 𝑖𝑛 3𝑥−1=4𝑥+2 Solution: 3𝑥−1=4𝑥+2 → use cross multiplication 3(𝑥+2)=4(𝑥−1) → use distributive property 3𝑥+6=4𝑥−4 → use addition property of equality 3𝑥−4𝑥=−4−6 → perform the operations. −𝑥=−10 → divide both sides by -1 Hence, 𝒙=𝟏𝟎

25. Example 2: Solve for 𝑥 𝑖𝑛 45=3𝑥−1+12 Solution: 45=3𝑥−1+12 → LCD is 10(x-1) 45∙10(𝑥−1)=3𝑥−1∙10(𝑥−1)+12∙10(𝑥−1) → perform the operations 4∙2(𝑥−1)=3(10)+5(𝑥−1) → distributive property 8𝑥−8=30+5𝑥−5 → combine like terms and solve for x 8𝑥−5𝑥=30−5+8 → simplify 3𝑥=33 → divide both sides by 3 𝑥=11 Hence, 𝒙=𝟏𝟏

26. The critical values are simply the zeros of both the numerator and the denominator. You must remember that the zeros of the denominator make the rational expression undefined, so they must be immediately disregarded or excluded as a possible solution. However, zeros of the numerator also need to be checked for its possible inclusion to the overall solution. Example: Solve the rational inequality below.

27. NOW IT’S YOUR TURN! Solve the following rational equations and inequality. Show your solutions and write your answers on a separate sheet of paper.

28. Remember

29. Representations of Rational Functions Lesson 8

30. How do we represent rational functions through an equation? In mathematics, a rational function is any function which can be defined by a rational fraction, for example, an algebraic fraction such that both the numerator and the denominator are polynomials. The denominator should not be equal to zero also. Note that f(x) is just the same as y.

31. NOW IT’S YOUR TURN!

32. Remember o Rational functions can be represented through equations, table of values and graphs.

33. Domain and Range of Rational Functions Lesson 9

34. REVIEW These are the terms or group of terms you need to know before going to the discussion on the domain and range of rational functions. 1. Set of Real Numbers (ℝ) – The real numbers include natural numbers or counting numbers, whole numbers, integers, rational numbers (fractions and repeating or terminating decimals), and irrational numbers. The set of real numbers consists of all the numbers that have a location on the number line. 2. Domain – the set of all x – values in a relation. 3. Range – the set of all y – values in a relation.

35. 4. Degree of Polynomial – the degree of a polynomial with one variable is based on the highest exponent. For example, in the expression x3 + 2x + 1, the degree is 3 since the highest exponent is 3.

36. The domain of the rational function 𝑓 is the set of real numbers except those values of x that will make the denominator zero.

37. How to find the Domain of a Rational Function? The domain of a function consists of the set of all real number (ℝ) except the value(s) that make the denominator zero.

38. How to find the Range of a Rational Function? The range of a rational function f is the set of real numbers except those values that fall to the following conditions.

39. NOW IT’S YOUR TURN!

40. Remember The domain of the rational function 𝑓 is the set of real numbers except those values of x that will make the denominator zero. • The range of a rational function f is the set of real numbers except those values that fall to the following conditions. Case 1: Same degree in the numerator and denominator Case 2: Numerator has a lower degree than the denominator

41. Assessment (Post-test)

42. Test I. Choose the letter of the correct answer and write them on a separate sheet of paper. 1. Melvin can finish a job in 9 hours working alone. Vanessa has more experience and can finish the same job in 6 hours working alone. How long will it take both people to finish that job working together? A. 2.3 hours B. 2.9 hours C. 3.6 hours D. 3.9 hours 2. Liza can finish a job in 5 hours working alone. If Liza and Enrique work together, they can finish the work in 3 hours. How long will it take if Enrique will choose to work alone? A. 10 hours B. 7.5 hours C. 7 hours D. 6.5 hours