LAS1_MATH_Week-1_Q2.pdf

Department of Education Division of Negros Occidental NEGROS OCCIDENTAL HIGH SCHOOL Mathematics Department Mathematics 8 Learning Activity Sheet #1 2nd QUARTER – Week 1 1 Written by: ROSE JEAN A...

Department of Education Division of Negros Occidental NEGROS OCCIDENTAL HIGH SCHOOL Mathematics Department Mathematics 8 Learning Activity Sheet #1 2nd QUARTER – Week 1 1 Written by: ROSE JEAN A. LOBEROS, NOHS 12/2020  NEGROS OCCIDENTAL HIGH SCHOOL MATHEMATICS 8 Learning Activity Sheet #1 2nd QUARTER - WEEK 1 I. LEARNING COMPETENCY: Differentiates Linear Inequalities in Two Variables from Linear Equations in Two Variables (M8AL-IIa-2) Illustrates and graphs Linear Inequalities in Two Variables (M8AL-IIa-1&3). II. BACKGROUND LINEAR EQUATION IN TWO VARIABLES is an equation of degree one (1) that is written in the form 𝐴𝑥 + 𝐵𝑦 = 𝐶, where A, B and C are real numbers and A and B are not both equal to zero. LINEAR INEQUALITIES IN TWO VARIABLES is an inequality that can be written in one of the following forms: 𝑨𝒙 + 𝑩𝒚 < 𝐶 , 𝑨𝒙 + 𝑩𝒚 > 𝐶 , 𝑨𝒙 + 𝑩𝒚 ≤ 𝑪 or 𝑨𝒙 + 𝑩𝒚 ≥ 𝑪 where A, B, and C are real numbers and A and B are not both equal to zero. Linear inequalities differ from linear equation, instead of using equal sign linear inequalities uses (greater than), ≤ (𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜), 𝑎𝑛𝑑 ≥ ( 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜). 2 Written by: ROSE JEAN A. LOBEROS, NOHS 12/2020  The graph of the linear inequalities in two variables is a set of all points in the rectangular coordinate system whose ordered pairs satisfy the inequality. When a line is graphed in the coordinate plane, it separates the plane into two regions called half –planes. The line that separates the plane which is the boundary of the graph is called the plane divider. To graph linear Inequalities in two variables: 1. Rewrite the inequality so that only “y” is on the left of the inequality sign and everything else on the right. 2. Graph the boundary line (by assigning values for x and solve for y). Make it a solid line for “𝒚 ≤ ”, and “𝒚 ≥ ”, then broken or dashed line for “𝒚 < ” and “𝒚 > ”. 3. Shade above the plane divider for “greater than”(𝑦 > 𝑜𝑟 𝑦 ≥ ), while shade below the plane divider for “less than” (𝑦 < 𝑜𝑟 𝑦 ≤). The shaded area indicates all the solution set in a given linear inequalities. III. REFERENCES AND RESOURCES Mathematics 8 Learner’s Module Unit 4 Mathematics 8 Teacher’s Guide Unit 4 https://math.tutorvista.com/algebra/linear-inequalities-in- two-variables.html http://www.algebra-class.com/graphing-inequalities.html http://www.kgsepg.com/project-id/6565-inequalities-two- variables 3 Written by: ROSE JEAN A. LOBEROS, NOHS 12/2020 IV. ACTIVITY Example 1: Determine whether the given below is a Linear Equation in Two Variables or Linear Inequalities in Two Variables. 1) −6𝑥 = 4 + 2𝑦 Linear Equation in Two Variables 2) 𝑥 + 8𝑦 = −4 Linear Equation in Two Variables 3) 2𝑥 − 3𝑦 ≥ 9 Linear Inequalities in Two Variables 4) 10𝑥 + 3𝑦 = 0 Linear Equation in Two Variables 5) 4 ≤ 2𝑦 − 11𝑥 Linear Inequalities in Two Variables 6) 𝑦 > −𝑥 − 5 Linear Inequalities in Two Variables 7) 𝑦 = 2𝑥 + 7𝑦 Linear Equation in Two Variables 8) 3𝑦 + 6𝑥 < 0 Linear Inequalities in Two Variables 9) 15𝑥 − 9 = 4𝑦 Linear Equation in Two Variables 3 1 10) 𝑥 + 𝑦 ≥ Linear Inequalities in Two Variables 4 2 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES: Example 2: Graph 𝒚 ≤ 𝟐𝒙 – 𝟏 Step 1: Rewrite the inequality so that only “y” is on the left of the inequality sign and everything else on the right. The inequality 𝒚 ≤ 𝟐𝒙 – 𝟏, has already “𝑦” on the left side of the inequality symbol, so no need to rewrite. Step 2: Graph the plane divider or boundary line. a. Assign values for 𝑥 and solve for 𝑦 𝑦 = 2𝑥 − 1 𝑦 = 2𝑥 − 1 𝑦 = 2𝑥 − 1 𝑦 = 2(0) − 1 𝑦 = 2(1) − 1 𝑦 = 2(2) − 1 𝑦 = −1 𝑦=1 𝑦=3 (0, −1) (1, 1) (2,3) 4 Written by: ROSE JEAN A. LOBEROS, NOHS 12/2020  b. Plot the points (0,-1), (1,1), (2,3) c. Connect with a solid line because of the inequality symbol (𝟎, −𝟏) (𝟏, 𝟏) (𝟐, 𝟑) Step 3. 3: Shade the area below below (because the planeydivideris “𝑦 ≤ because“𝒚 ”). ≤ ". The shaded region are the solutions in the inequality y ≤ 2x – 1. Example (2,3), The points (2, (4,1), 3), (4, 1), (3,(3,2) 2) areare solutions some of the because they are solutions, because they are located in the shaded region. in the shaded region, while the located points (-1,3), Points (-1, 3), (-1,(-1,5), 5), (0, 4) are not solutions of the (0,4) are not solutions. given inequality, because they are not in the shaded region. Example 3: Graph 2𝑦 – 𝑥 ≤ 6 Step 1: Rewrite the inequality so that only “y” is on the left of the inequality sign and everything else on the right. 2𝑦 – 𝑥 ≤ 6 2𝑦 − 𝑥 + 𝑥 ≤ 𝑥 + 6 (add x to both sides to eliminate −𝑥 in the left side) 2y ≤ x + 6 2𝑦 𝑥 6 2 ≤ 2 + 2 (divide both sides by 2 to eliminate 2 in the left side) x y≤2+3 5 Written by: ROSE JEAN A. LOBEROS, NOHS 12/2020 Step 2: Graph the plane divider or boundary line. a. Assign values for x and solve for y (0, 3) (2, 4) b. Plot the points (-2, 2), (0, 3), (2,4) c. Connect with a solid line because of the inequality symbol ≤. 𝒙 𝒚≤ +𝟑 𝟐 (−𝟐, 𝟐) (𝟎, 𝟑) (𝟐, 𝟒) Step 3: Shade the area below the plane divider because“𝒚 ≤ ". Step 3. Shade the area below (because “y ≤”): The shaded region are the solutions in the 𝒙 The shaded inequality 2𝑦region − 𝑥 ≤are the 6 or 𝒚≤solutions 𝟐 + 𝟑. in 𝑥 thepoints The inequality 𝑦3),≤(4,2 2), (1, 1), (2, + (0,3), 3. (2,4)Example are some of the solutions, because they are located in the shaded (1,1), (2,3), (4,2) are solutions because region and the boundary is a solid line. they are Points (3, 6), (-1, 5), (0, 6) are not solutions of the located given in thebecause inequality, shaded region, they are not while the in the shaded points (3,6), (-1,5), region. (0,6) are6not solutions. Written by: ROSE JEAN A. LOBEROS, NOHS 12/2020  𝒚 Example 4: Graph + 𝟐 > 𝑥 𝟐 Step 1: Rewrite the inequality so that only “y” is on the left of the inequality sign and everything else on the right. 𝑦 + 2 > 𝑥 2 𝑦 + 2−2 > 𝑥−2 (subtract 2 to both sides to eliminate 2 in the left) 2 𝑦 2 [ 2 > 𝑥 − 2] (multiply both sides by 2) 𝑦 > 2𝑥 − 4 Step 2: Graph the plane divider or boundary line. a. Assign values for x and solve for y b. Plot the points (1,-2), (2, 0), (3,2) c. Connect the points with a broken line or dashed line because “𝒚 > ”, which points on the line is not included in the solutions. 7 Written by: ROSE JEAN A. LOBEROS, NOHS 12/2020  Step 3: Shade the area above the plane divider because “𝒚 >”. The Theshaded shadedregion areare region thethe solutions in solutions 𝒚 the inequality+𝑦𝟐 >> 2𝑥 inequality in the 𝟐 𝑥 or 𝑦 > 2𝑥 − – 4. 4, and the points on the boundary are Example (1,1), (2,3), (1,2) are not a solutions. solutions because they are The points in located (1, the 1), (2,shaded 3), (1, 2) are some ofwhile region, the solutions, the points because they are (3, −2), located (4,1), (3,in−2) the shaded are region. not solutions. Points (3, -2), (4, 1), (3, 2) are not solutions of the given inequality, because they are not in the shaded region. Practice Exercises (For Review and Study purposes. DO NOT SUBMIT THE ANSWERS.) A. Determine whether the following is a Linear Equation in Two Variables or Linear Inequality in Two Variables. 1) 5𝑥 − 𝑦 < 2 6) 𝑦 ≥ 9𝑥 + 3 2 2) −2𝑥 = 4 + 𝑦 7) 3 𝑥 − 𝑦 = −2 3) 10𝑥 + 5 = 𝑦 8) 𝑥 − 7𝑦 < 2 4) 𝑥 ≤ 3𝑦 + 5 9) 12𝑥 ≤ −𝑦 + 6 5) 8𝑦 ≥ 3 + 𝑥 10) 𝑥 + 𝑦 = 8 B. Graph the following. 1. y ≤ x + 2 2. 2y + 4x > −8 8 Written by: ROSE JEAN A. LOBEROS, NOHS 12/2020 V. ADDITIONAL ACTIVITY:(For Study Purposes. DO NOT SUBMIT THE ANSWERS.) Direction: Choose the letter of the BEST answer. 1. What is an inequality that can be written in any of the following forms:𝑨𝒙 + 𝑩𝒚 < 𝐶 , 𝑨𝒙 + 𝑩𝒚 > 𝐶, 𝑨𝒙 + 𝑩𝒚 ≤ 𝑪, or 𝑨𝒙 + 𝑩𝒚 ≥ 𝑪, where A, B, and C are real numbers and A and B are not both equal to zero? A. Exponential Equation C. Linear Inequality B. Linear Equation D. Quadratic Equation 2. Which of the following is a Linear Inequality in Two Variables? A. 7𝑥 – 4𝑦 > 3 C. 2𝑦 ≤ 6 B. 4𝑥 + 3 < 23 D. −5𝑦 ≥ 3 3. Which of the following best describe the graph of Linear Inequalities in Two Variables? A. Curve Line C. Parabola B. Half Plane D. Straight Line 4. A Linear Inequality in Two Variables has ________ solution/s. A. 0 B. 1 C. 2 D. Infinite 5. What do you call the line that separate the plane into two regions? A. Equalizer C. Plane Divider B. Half Plane D. Quadrants 6. Which of the following is NOT a Linear Inequality in Two Variables? A. 4𝑎 − 2𝑏 > 7 C. 6𝑝 ≥ 2 B. 4 + 8𝑥 < 14𝑦 D. 5𝑚 + 6 > 3𝑛 7. When can you say that an inequality will have a solid line as a plane divider? A. The symbol is equal sign C. The symbol is greater than B. The symbol has “or equal to” D. The symbol is lesser than 9 Written by: ROSE JEAN A. LOBEROS, NOHS 12/2020 8. Which of the following is NOT a Linear Inequality in Two Variables? A. 3𝑥 + 2 > 2𝑦 C. 7 + 4𝑟 < 3𝑡 B. 5𝑥 2 ≥ 3𝑦 + 1 D. 4𝑎 − 2𝑏 > 7 9. What type of line and where will you shade with the sign “ −8 9. A 2𝑦 > −4𝑥 − 8 10. B 2𝑦 2 > −4𝑥 2 − 82 𝑦 > −2𝑥 − 4 11 Written by: ROSE JEAN A. LOBEROS, NOHS 12/2020