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**Solving Systems of Inequalities Graphically** Below is a task that requires solving a system of inequalities graphically and identifying the coordinates of a point within the solution set. ### Inequalities: 1. \( y \geq -\frac{3}{4}x + 1 \) 2. \( y \geq \frac{1}{2}x + 6 \) ### Instructions: Use the coordinate axes provided to graph the two inequalities. The area of overlap among the regions satisfying both inequalities is the solution set. ### Graph Description: There is a coordinate plane with the `x-axis` labeled from -10 to 10 and the `y-axis` also labeled from -1 to 10. 1. **First Inequality**: \( y \geq -\frac{3}{4}x + 1 \) - This line has a y-intercept at \( (0, 1) \). - The slope of the line is -3/4, which means for every increase of 4 units in `x`, `y` decreases by 3 units. - The line is solid because the inequality includes \( y = -\frac{3}{4}x + 1 \). - The region above this line is shaded to represent the solution set for this inequality. 2. **Second Inequality**: \( y \geq \frac{1}{2}x + 6 \) - This line has a y-intercept at \( (0, 6) \). - The slope of the line is 1/2, which means for every increase of 2 units in `x`, `y` increases by 1 unit. - The line is solid because the inequality includes \( y = \frac{1}{2}x + 6 \). - The region above this line is shaded to represent the solution set for this inequality. ### Solution Set: The solution set for the system of inequalities is the region where the shading of both inequalities overlaps. To determine a specific point within this overlap, identify any coordinate that lies within this shared region. For example, a possible coordinate within the solution set can be visually identified on the graph. ### Conclusion: By graphing the given inequalities and identifying the overlapping region, we can determine the solution set for the system. Any point within this region satisfies both inequalities.

**Solving Systems of Inequalities Graphically** Below is a task that requires solving a system of inequalities graphically and identifying the coordinates of a point within the solution set. ### Inequalities: 1. \( y \geq -\frac{3}{4}x + 1 \) 2. \( y \geq \frac{1}{2}x + 6 \) ### Instructions: Use the coordinate axes provided to graph the two inequalities. The area of overlap among the regions satisfying both inequalities is the solution set. ### Graph Description: There is a coordinate plane with the `x-axis` labeled from -10 to 10 and the `y-axis` also labeled from -1 to 10. 1. **First Inequality**: \( y \geq -\frac{3}{4}x + 1 \) - This line has a y-intercept at \( (0, 1) \). - The slope of the line is -3/4, which means for every increase of 4 units in `x`, `y` decreases by 3 units. - The line is solid because the inequality includes \( y = -\frac{3}{4}x + 1 \). - The region above this line is shaded to represent the solution set for this inequality. 2. **Second Inequality**: \( y \geq \frac{1}{2}x + 6 \) - This line has a y-intercept at \( (0, 6) \). - The slope of the line is 1/2, which means for every increase of 2 units in `x`, `y` increases by 1 unit. - The line is solid because the inequality includes \( y = \frac{1}{2}x + 6 \). - The region above this line is shaded to represent the solution set for this inequality. ### Solution Set: The solution set for the system of inequalities is the region where the shading of both inequalities overlaps. To determine a specific point within this overlap, identify any coordinate that lies within this shared region. For example, a possible coordinate within the solution set can be visually identified on the graph. ### Conclusion: By graphing the given inequalities and identifying the overlapping region, we can determine the solution set for the system. Any point within this region satisfies both inequalities.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Solving Systems of Inequalities Graphically** Below is a task that requires solving a system of inequalities graphically and identifying the coordinates of a point within the solution set. ### Inequalities: 1. \( y \geq -\frac{3}{4}x + 1 \) 2. \( y \geq \frac{1}{2}x + 6 \) ### Instructions: Use the coordinate axes provided to graph the two inequalities. The area of overlap among the regions satisfying both inequalities is the solution set. ### Graph Description: There is a coordinate plane with the `x-axis` labeled from -10 to 10 and the `y-axis` also labeled from -1 to 10. 1. **First Inequality**: \( y \geq -\frac{3}{4}x + 1 \) - This line has a y-intercept at \( (0, 1) \). - The slope of the line is -3/4, which means for every increase of 4 units in `x`, `y` decreases by 3 units. - The line is solid because the inequality includes \( y = -\frac{3}{4}x + 1 \). - The region above this line is shaded to represent the solution set for this inequality. 2. **Second Inequality**: \( y \geq \frac{1}{2}x + 6 \) - This line has a y-intercept at \( (0, 6) \). - The slope of the line is 1/2, which means for every increase of 2 units in `x`, `y` increases by 1 unit. - The line is solid because the inequality includes \( y = \frac{1}{2}x + 6 \). - The region above this line is shaded to represent the solution set for this inequality. ### Solution Set: The solution set for the system of inequalities is the region where the shading of both inequalities overlaps. To determine a specific point within this overlap, identify any coordinate that lies within this shared region. For example, a possible coordinate within the solution set can be visually identified on the graph. ### Conclusion: By graphing the given inequalities and identifying the overlapping region, we can determine the solution set for the system. Any point within this region satisfies both inequalities.
Transcribed Image Text:**Solving Systems of Inequalities Graphically** Below is a task that requires solving a system of inequalities graphically and identifying the coordinates of a point within the solution set. ### Inequalities: 1. \( y \geq -\frac{3}{4}x + 1 \) 2. \( y \geq \frac{1}{2}x + 6 \) ### Instructions: Use the coordinate axes provided to graph the two inequalities. The area of overlap among the regions satisfying both inequalities is the solution set. ### Graph Description: There is a coordinate plane with the `x-axis` labeled from -10 to 10 and the `y-axis` also labeled from -1 to 10. 1. **First Inequality**: \( y \geq -\frac{3}{4}x + 1 \) - This line has a y-intercept at \( (0, 1) \). - The slope of the line is -3/4, which means for every increase of 4 units in `x`, `y` decreases by 3 units. - The line is solid because the inequality includes \( y = -\frac{3}{4}x + 1 \). - The region above this line is shaded to represent the solution set for this inequality. 2. **Second Inequality**: \( y \geq \frac{1}{2}x + 6 \) - This line has a y-intercept at \( (0, 6) \). - The slope of the line is 1/2, which means for every increase of 2 units in `x`, `y` increases by 1 unit. - The line is solid because the inequality includes \( y = \frac{1}{2}x + 6 \). - The region above this line is shaded to represent the solution set for this inequality. ### Solution Set: The solution set for the system of inequalities is the region where the shading of both inequalities overlaps. To determine a specific point within this overlap, identify any coordinate that lies within this shared region. For example, a possible coordinate within the solution set can be visually identified on the graph. ### Conclusion: By graphing the given inequalities and identifying the overlapping region, we can determine the solution set for the system. Any point within this region satisfies both inequalities.
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