4.5 Systems of Inequalities

Front

What is a system of inequalities?

Back

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

Front

How do you graph a system of inequalities?

Back

To graph a system of inequalities, first graph each inequality as if it were an equation. Use a dashed line for < or > and a solid line for ≤ or ≥. Then, shade the appropriate region for each inequality, and the solution is where the shaded regions overlap.

Front

What does it mean for a point to be a solution to a system of inequalities?

Back

A point is a solution to a system of inequalities if it satisfies all inequalities in the system, meaning it lies in the overlapping shaded region of the graph.

Front

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

Back

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

Front

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

Back

To determine if a point is a solution, substitute the x and y values of the point into each inequality. If the point satisfies all inequalities, it is a solution.

Front

What is the solution set of a system of inequalities?

Back

The solution set is the collection of all points that satisfy all inequalities in the system, often represented as a shaded region on a graph.

Front

What does the inequality y > -2x + 3 represent graphically?

Back

It represents the region above the line y = -2x + 3, where the line itself is dashed, indicating that points on the line are not included in the solution.

Front

What is the significance of the ordered pair (2, 4) in a system of inequalities?

Back

The ordered pair (2, 4) is a solution if it satisfies all inequalities in the system, meaning it lies within the overlapping shaded region.

Front

What does it mean if a point is not a solution to a system of inequalities?

Back

If a point is not a solution, it means that when substituted into the inequalities, it does not satisfy at least one of them, and thus does not lie in the solution set.

Front

How can you write an inequality from a word problem?

Back

Identify the quantities involved, determine the relationship (greater than, less than, etc.), and express it using inequality symbols.