13 2 13 10 13 13 = 13 16 13 –3 Therefore Therefore is Therefore 5 is Therefore 8 is Therefore 11 is –8 –3 is not a not a solution. not a solution. a solution. not a solution. solution. Based on the evaluation, only x = 8 satisfied the equation while the rest did not. Therefore, we proved that only one element in the replacement set satisfies the equation. We can also use a similar procedure to find solutions to a mathematical inequality, as the following example shows. Example 2. Given, x 3 < 5, determine the element/s of the replacement set – {–8,–3, 5, 8, 11} that satisfy the inequality. x 3 < 5 – For x = For x = For x = 5: For x = 8: For x = 11: –8: –3: 3 = 3 = 5 3 = 2 8 3 = 5 11 3 = 8 –8 – –11 –3 – –6 – – – < 5 < 5 2 < 5 5 < 5 8 < 13 –11 –6 Therefore Therefore is Therefore 5 is Therefore 8 is Therefore 11 is –3 -8 is a a solution. a solution. a solution. not a solution. solution. Based on the evaluation, the inequality was satisfied if x = 5, or 8. The –8,–3, inequality was not satisfied when x = 11. Therefore, there are 4 elements in the replacement set that are solutions to the inequality. VI. Exercises Given the replacement set {–3, 0, 1, 2, 3}, determine the solution/s for –2, –1, the following equations and inequalities. Show your step-by-step computations to prove your conclusion. x + 8 < 10 2x + 4 = 3 x 5 > 3 – – x > 4 and x < 2 – x < 0 and x > 2.5 Solve for the value of x to make the mathematical sentence true. You may try several values for x until you reach a correct solution. x + 6 = 10 x 4 = 11 – 2x = 8
–3 13 Therefore –8 is not a solution. 2 13 Therefore –3 is not a solution. 10 13 Therefore 5 is not a solution. 13 = 13 Therefore 8 is a solution. 16 13 Therefore 11 is not a solution. Based on the evaluation, only x = 8 satisfied the equation while the rest did not. Therefore, we proved that only one element in the replacement set satisfies the equation. We can also use a similar procedure to find solutions to a mathematical inequality, as the following example shows. Example 2. Given, x – 3 < 5, determine the element/s of the replacement set {–8,–3, 5, 8, 11} that satisfy the inequality. x – 3 < 5 For x = –8: –8 – 3 = –11 –11 < 5 Therefore -8 is a solution. For x = –3: –3 – 3 = –6 –6 < 5 Therefore –3 is a solution. For x = 5: 5 – 3 = 2 2 < 5 Therefore 5 is a solution. For x = 8: 8 – 3 = 5 5 < 5 Therefore 8 is a solution. For x = 11: 11 – 3 = 8 8 < 13 Therefore 11 is not a solution. Based on the evaluation, the inequality was satisfied if x = –8,–3, 5, or 8. The inequality was not satisfied when x = 11. Therefore, there are 4 elements in the replacement set that are solutions to the inequality. VI. Exercises Given the replacement set {–3, –2, –1, 0, 1, 2, 3}, determine the solution/s for the following equations and inequalities. Show your step-by-step computations to prove your conclusion. x + 8 < 10 2x + 4 = 3 x – 5 > – 3 x > – 4 and x < 2 x < 0 and x > 2.5 Solve for the value of x to make the mathematical sentence true. You may try several values for x until you reach a correct solution. x + 6 = 10 x – 4 = 11 2x = 8