Solutions to 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 ordered pairs that satisfy all inequalities in the system.

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 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 the x and y values are substituted into the inequalities, at least one inequality is not satisfied.

Front

What is the graphical representation of a system of inequalities?

Back

The graphical representation consists of the shaded regions of the inequalities on a coordinate plane, where the overlapping shaded area represents the solution set.

Front

What does the inequality y < 3 represent graphically?

Back

The inequality y < 3 represents all points below the horizontal line y = 3, not including the line itself.

Front

How can you identify a solution from a set of ordered pairs?

Back

Substitute each ordered pair into the inequalities of the system. The pair that satisfies all inequalities is the solution.

Front

What is the significance of the boundary line in a system of inequalities?

Back

The boundary line represents the points where the inequality changes from true to false. For strict inequalities (< or >), the line is dashed; for inclusive inequalities (≤ or ≥), the line is solid.

Front

What does it mean for a point to be on the boundary line of an inequality?

Back

A point on the boundary line satisfies the equation of the line but may or may not satisfy the inequality, depending on whether the inequality is strict or inclusive.

Front

How do you graph the inequality y ≥ 2x + 1?

Back

First, graph the line y = 2x + 1 as a solid line. Then, shade the region above the line to represent all points where y is greater than or equal to 2x + 1.

Front

What is the solution set of a system of inequalities?

Back

The solution set is the collection of all ordered pairs that satisfy all inequalities in the system, represented graphically by the overlapping shaded regions.