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Answer :
Final answer:
The solutions to the inequality on the coordinate plane are points c) (1, 1) and e) (2, 5), as they fall within the shaded area to the right of the line with the equation 'y = 2x + 3'.
Explanation:
Understanding Coordinate Planes and Inequalities
To determine which points are solutions to the inequality represented by the shaded region on a coordinate plane, we need to consider the equation of the line and the direction of the shading. Since the line goes through (-1.5, 0) and (0, 3) with a positive slope and the shading is to the right, the inequality this line represents is 'y > mx + b', where 'm' is the slope and 'b' is the y-intercept.
By comparing the slope between the two given points, we can find that the slope 'm' is 2, because going from (-1.5, 0) to (0, 3) we move 1.5 units to the right (positive direction on the x-axis) and 3 units up (positive direction on the y-axis), so the slope is 3/1.5 or 2. The y-intercept is the point where the line crosses the y-axis, which is, in this case, 3. Therefore, the equation of the line is 'y = 2x + 3'.
Let's test the given points:
- (3, 3) would not be a solution because plugging x = 3 into the equation gives us y = 2(3) + 3 = 9, which is not less than 3.
- (2, 2) would not be a solution for the same reason, as the y-value of the line at x = 2 is 7, which is greater than 2.
- (1, 1) is indeed a solution because when x = 1, the equation yields y = 2(1) + 3 = 5, and 1 is less than 5.
- (0, 1) would not be a solution because the y-value of the line at x = 0 is 3, which is greater than 1.
- (2, 5) is also a solution as plugging x = 2 into the equation gives y = 2(2) + 3 = 7, and 5 is less than 7.
- Therefore, the points that satisfy the inequality represented by the shaded region are (1, 1) and (2, 5).
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