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6) Determine if each point is a solution to the system of inequalities shown. STO (1, 1) (-3, 4) (3,5) (4,3) (1,-5) (-5, -1) 7) Determine the system of inequalities that is graphed to the right. 1 1 -1 1 1 1 y 4 3 8) Use your answer to problem 7 to prove that the point (24, 11) is a solution to the system of equations shown in the graph.

6) Determine if each point is a solution to the system of inequalities shown. STO (1, 1) (-3, 4) (3,5) (4,3) (1,-5) (-5, -1) 7) Determine the system of inequalities that is graphed to the right. 1 1 -1 1 1 1 y 4 3 8) Use your answer to problem 7 to prove that the point (24, 11) is a solution to the system of equations shown in the graph.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Educational Content Transcription** ### Exercises on Systems of Inequalities **6) Determine if each point is a solution to the system of inequalities shown.** - Points to check: - (1, 1) - (4, 3) - (-3, 4) - (1, -5) - (3, 5) - (-5, -1) **7) Determine the system of inequalities that is graphed to the right.** *Graph Explanation:* - The graph features two inequalities. One is a linear inequality represented by a solid line, which slopes downwards as it moves from left to right. The shaded region for this inequality lies beneath this line. - The second inequality appears as a parabolic curve, opening upwards. The shaded region is inside the curve, beneath the parabola and excluding the exterior area. **8) Use your answer to problem 7 to prove that the point (24, 11) is a solution to the system of equations shown in the graph.** **9) Graph these non-linear inequalities:** - \( y < -(x + 3)^2 + 3 \) *Graph Explanation:* The inequality features a downward-opening parabola shifted 3 units to the left, and 3 units up from the origin. The region of interest is below the parabola. - \( y \geq \frac{1}{2} |x - 2| - 3 \) *Graph Explanation:* This inequality represents an absolute value function, creating a V-shape. The vertex is located at (2, -3). The region of interest is above and including the V-shaped line.
Transcribed Image Text:**Educational Content Transcription** ### Exercises on Systems of Inequalities **6) Determine if each point is a solution to the system of inequalities shown.** - Points to check: - (1, 1) - (4, 3) - (-3, 4) - (1, -5) - (3, 5) - (-5, -1) **7) Determine the system of inequalities that is graphed to the right.** *Graph Explanation:* - The graph features two inequalities. One is a linear inequality represented by a solid line, which slopes downwards as it moves from left to right. The shaded region for this inequality lies beneath this line. - The second inequality appears as a parabolic curve, opening upwards. The shaded region is inside the curve, beneath the parabola and excluding the exterior area. **8) Use your answer to problem 7 to prove that the point (24, 11) is a solution to the system of equations shown in the graph.** **9) Graph these non-linear inequalities:** - \( y < -(x + 3)^2 + 3 \) *Graph Explanation:* The inequality features a downward-opening parabola shifted 3 units to the left, and 3 units up from the origin. The region of interest is below the parabola. - \( y \geq \frac{1}{2} |x - 2| - 3 \) *Graph Explanation:* This inequality represents an absolute value function, creating a V-shape. The vertex is located at (2, -3). The region of interest is above and including the V-shaped line.
## Understanding Graphical and Algebraic Solutions of Inequalities ### Problem 1: Solving Inequality Algebraically Given the inequality \(4x + 5y \geq 12\), determine if each point is a solution: - **a. (-2, 4)** - **b. (3, -2)** - **c. (0, 2)** - **d. (2, 5)** ### Problem 2: Graphing the Inequality Graph the inequality from Problem 1 on the coordinate plane provided. ### Problem 3: Verifying Solutions Using the Graph Use the graph to check if your answers to Problem 2 are correct. ### Problem 4: Checking Additional Points Use the graph to determine if the following points are solutions to \(4x + 5y \geq 12\): - **a. (3.7, 2.15)** - **b. (-2, 4.88)** - **c. (5.6, -3.99)** - **d. (0, 0)** ### Problem 5: Graphing Additional Inequalities Graph each inequality and pay attention to the type of boundary line: - **a) \(|x| \geq 2\)** - **b) \(y > -(x+2)^2 + 3\)** ### Graphical Explanations **Graph for Problem 2:** - The graph provided has a coordinate plane with axes ranging from -5 to 5 on both the x and y axes. The line representing \(4x + 5y = 12\) should be plotted on this coordinate plane. **Graph for Problem 5(a):** - This graph needs to represent the inequality \(|x| \geq 2\). The boundary lines should be vertical lines at \(x = -2\) and \(x = 2\) with shaded regions on the outside to indicate the solution set. **Graph for Problem 5(b):** - Graph \(y > -(x+2)^2 + 3\), which is a parabola opening downwards. The vertex of the parabola is at \((-2, 3)\). The solution area is above this curve, indicated by a dotted line for the inequality. These tasks are designed to enhance understanding of algebraic and
Transcribed Image Text:## Understanding Graphical and Algebraic Solutions of Inequalities ### Problem 1: Solving Inequality Algebraically Given the inequality \(4x + 5y \geq 12\), determine if each point is a solution: - **a. (-2, 4)** - **b. (3, -2)** - **c. (0, 2)** - **d. (2, 5)** ### Problem 2: Graphing the Inequality Graph the inequality from Problem 1 on the coordinate plane provided. ### Problem 3: Verifying Solutions Using the Graph Use the graph to check if your answers to Problem 2 are correct. ### Problem 4: Checking Additional Points Use the graph to determine if the following points are solutions to \(4x + 5y \geq 12\): - **a. (3.7, 2.15)** - **b. (-2, 4.88)** - **c. (5.6, -3.99)** - **d. (0, 0)** ### Problem 5: Graphing Additional Inequalities Graph each inequality and pay attention to the type of boundary line: - **a) \(|x| \geq 2\)** - **b) \(y > -(x+2)^2 + 3\)** ### Graphical Explanations **Graph for Problem 2:** - The graph provided has a coordinate plane with axes ranging from -5 to 5 on both the x and y axes. The line representing \(4x + 5y = 12\) should be plotted on this coordinate plane. **Graph for Problem 5(a):** - This graph needs to represent the inequality \(|x| \geq 2\). The boundary lines should be vertical lines at \(x = -2\) and \(x = 2\) with shaded regions on the outside to indicate the solution set. **Graph for Problem 5(b):** - Graph \(y > -(x+2)^2 + 3\), which is a parabola opening downwards. The vertex of the parabola is at \((-2, 3)\). The solution area is above this curve, indicated by a dotted line for the inequality. These tasks are designed to enhance understanding of algebraic and
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