Question

inequality system solutions quiz
algebra 1 a is (bva) 25-26 / module 6: systems ...

1. michal makes the statement, \all systems of linear inequalities with the same slope and different y-intercepts have no solution.\ is his statement correct?

the statement is correct. parallel boundary lines never have overlapping shaded areas.
the statement is incorrect because parallel boundary lines can have overlapping shaded areas depending on the graph.
the statement is correct. systems of linear inequalities do not have solutions.
the statement is incorrect. the boundary lines with the same slope will eventually intersect and will have one solution.

Explanation:

To determine if Michal's statement is correct, we analyze systems of linear inequalities with the same slope (parallel lines) and different y - intercepts.

1. Recall the properties of linear inequalities and their graphs: Linear inequalities with the same slope have parallel boundary lines. The solution to a system of linear inequalities is the region where the shaded areas of the individual inequalities overlap.
2. Analyze the options:

• Option 1: Claims parallel boundary lines never have overlapping shaded areas. But this is not true. For example, consider \(y>2x + 1\) and \(y>2x-1\). The shaded regions (above each line) will overlap for \(y\) - values that are greater than both \(2x + 1\) and \(2x-1\) (since \(2x + 1>2x - 1\) for all \(x\), the region \(y>2x + 1\) is a subset of \(y>2x-1\), so there is an overlapping region).
• Option 2: States the statement is incorrect because parallel boundary lines can have overlapping shaded areas depending on the graph. This is correct. The direction of the inequality (e.g., \(y\geq mx + b\) or \(y\leq mx + b\)) and the position of the lines (different y - intercepts but same slope) determine if there is an overlapping region.
• Option 3: Says systems of linear inequalities do not have solutions. This is false. Many systems of linear inequalities do have solutions (e.g., \(y>x\) and \(y < x + 1\) has a solution region between the two lines).
• Option 4: Claims that boundary lines with the same slope will eventually intersect. But lines with the same slope are parallel and never intersect, so this is incorrect.

Answer:

B. The statement is incorrect because parallel boundary lines can have overlapping shaded areas depending on the graph.