Consider the following system of Inequalities
y 2x - 6
y < 5x + 2
Describe how you could determine whether if ( (5, 2) is a solution to the system of inequalities
To determine whether (5, 2) is a solution to the system of inequalities, we substitute the values of x and y into each inequality and check if the resulting statements are true.
For the first inequality, y ≥ 2x - 6:
Substituting x = 5 and y = 2, we have:
2 ≥ 2(5) - 6
2 ≥ 10 - 6
2 ≥ 4
Since 2 is not greater than or equal to 4, the statement is false.
For the second inequality, y < 5x + 2:
Substituting x = 5 and y = 2, we have:
2 < 5(5) + 2
2 < 25 + 2
2 < 27
Since 2 is less than 27, the statement is true.
Therefore, (5, 2) is not a solution to the system of inequalities because it does not satisfy the first inequality.
Could a system of linear inequalities ever have no solutions? What would this look like on a graph? Explain.
Yes, a system of linear inequalities can have no solutions. This occurs when the graph of the inequalities does not intersect, meaning there is no region that satisfies all the inequalities simultaneously.
On a graph, a system of linear inequalities without a solution would appear as parallel lines or lines that are parallel in some sections and do not intersect in any common region.
For example, consider the system of inequalities:
y < 2x + 3
y > 2x + 5
The graph of the first inequality would be a dashed line with a slope of 2 and a y-intercept of 3, as y is strictly less than 2x + 3. The graph of the second inequality would be another dashed line with the same slope but a y-intercept of 5, as y is strictly greater than 2x + 5.
Since the lines have the same slope and are parallel, they will not intersect. Thus, there is no region that satisfies both inequalities simultaneously. Therefore, this system of linear inequalities has no solutions.