Graph the equation y=4x+1 by plotting points.
2 answers:
Look at the attached image (credit to Desmos graphing calculator)
To graph the equation y=4x+1, we just find two points that exist, plot them, and draw a straight line that goes through them.
Let's try x=0:
One point on the line is (0,1)
Let's try x=1:
Another point is (1,5).
Now, plot these two points and draw a line through them.
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The red line is represented by y ≥ -3x + 3.
The blue line is represented by y < 3/2x - 6.
A) In this system of inequalities, there is one solid line and one dashed line.
The solid line is the inequality y ≥ -3x + 3 because it has the symbol greater than or equal to. The "equal to" part of the symbol is important because it means that any coordinate located on the solid line is a solution.
However, the inequality y < 3/2x - 6 is a dashed line because it does not have "equal to." This makes the inequality a dashed line which means that all coordinates on it will not be a solution to the inequality.
In the inequality y ≥ -3x + 3, the shading is above the line. This means that all coordinates within the line and shaded area above the line will be a solution to the inequality.
In the inequality y < 3/2x - 6, the shading is below the line. This means that all coordinates within the shared area of the line will be a solution to the inequality. However, since y is not less than or EQUAL TO, all coordinates within the dashed line is not a solution to the inequality.
The solution area, where a coordinate is a solution to
both inequalities, is where the shaded areas of both inequalities overlap.
B) The point (-6, 3) is a not solution to both of the inequalities because it is not within the shared areas of the two inequalities.
Substitute the coordinate (-6, 3) in the inequality y < 3/2x - 6.
y < 3/2x - 6
3 < 3/2(-6) - 6
3 < -9 - 6
3 < -15
This is not true because a negative number cannot be greater than a positive number. Therefore, this proves that the coordinate (-6, 3) is not a solution for the inequality y < 3/2x - 6.
Check if the coordinate fits for the other inequality.
y ≥ -3x + 3
3 ≥ -3(-6) + 3
3 ≥ 18 + 3
3 ≥ 21
This is also not true because 3 is not greater than or equal to 21. Therefore, this coordinate is not a solution to the inequality y ≥ -3x + 3.
The coordinate (-6, 3) is not a solution for the system of inequalities because it is not a solution for both inequalities.
The blue line is represented by y < 3/2x - 6.
A) In this system of inequalities, there is one solid line and one dashed line.
The solid line is the inequality y ≥ -3x + 3 because it has the symbol greater than or equal to. The "equal to" part of the symbol is important because it means that any coordinate located on the solid line is a solution.
However, the inequality y < 3/2x - 6 is a dashed line because it does not have "equal to." This makes the inequality a dashed line which means that all coordinates on it will not be a solution to the inequality.
In the inequality y ≥ -3x + 3, the shading is above the line. This means that all coordinates within the line and shaded area above the line will be a solution to the inequality.
In the inequality y < 3/2x - 6, the shading is below the line. This means that all coordinates within the shared area of the line will be a solution to the inequality. However, since y is not less than or EQUAL TO, all coordinates within the dashed line is not a solution to the inequality.
The solution area, where a coordinate is a solution to
both inequalities, is where the shaded areas of both inequalities overlap.
B) The point (-6, 3) is a not solution to both of the inequalities because it is not within the shared areas of the two inequalities.
Substitute the coordinate (-6, 3) in the inequality y < 3/2x - 6.
y < 3/2x - 6
3 < 3/2(-6) - 6
3 < -9 - 6
3 < -15
This is not true because a negative number cannot be greater than a positive number. Therefore, this proves that the coordinate (-6, 3) is not a solution for the inequality y < 3/2x - 6.
Check if the coordinate fits for the other inequality.
y ≥ -3x + 3
3 ≥ -3(-6) + 3
3 ≥ 18 + 3
3 ≥ 21
This is also not true because 3 is not greater than or equal to 21. Therefore, this coordinate is not a solution to the inequality y ≥ -3x + 3.
The coordinate (-6, 3) is not a solution for the system of inequalities because it is not a solution for both inequalities.