Systems of Inequalities

Front

What is a system of inequalities?

Back

A system of inequalities is a set of two or more inequalities that share the same variables. The solution is the set of all points that satisfy all inequalities in the system.

Front

How do you graph a linear inequality?

Back

To graph a linear inequality, first graph the corresponding linear equation as a boundary line. Use a dashed line for < or > and a solid line for ≤ or ≥. Then, shade the region that satisfies the inequality.

Front

What does the solution set of a system of inequalities represent?

Back

The solution set of a system of inequalities represents all the points (x, y) that satisfy all inequalities in the system, often visualized as a shaded region on a graph.

Front

What is the difference between a solid line and a dashed line in graphing inequalities?

Back

A solid line indicates that points on the line are included in the solution (≤ or ≥), while a dashed line indicates that points on the line are not included ( < or > ).

Front

How can you determine if a point is a solution to a system of inequalities?

Back

To determine if a point is a solution, substitute the point's coordinates into each inequality. If the point satisfies all inequalities, it is part of the solution set.

Front

What does the inequality y ≤ 2x - 1 represent graphically?

Back

The inequality y ≤ 2x - 1 represents the region below or on the line y = 2x - 1, including the line itself.

Front

What is the significance of the slope in a linear inequality?

Back

The slope indicates the steepness of the line and the direction of the inequality. A positive slope means the line rises from left to right, while a negative slope means it falls.

Front

How do you find the intersection of two inequalities?

Back

To find the intersection, graph both inequalities and identify the region where the shaded areas overlap. This overlapping region represents the solution to the system.

Front

What does the inequality x < -1 represent?

Back

The inequality x < -1 represents all points to the left of the vertical line x = -1, not including the line itself.

Front

What is the role of boundary lines in systems of inequalities?

Back

Boundary lines separate the solution regions of inequalities. They help determine which side of the line satisfies the inequality.