Inequalities and Graphing
What does the solution to an inequality represent on a number line?
all values that make the inequality true.
What does it mean for an ordered pair to be a solution to a system of equations?
ordered pair satisfies both equations at the same time
What does a matrix allow us to do when working with systems of equations?
Matrices organize the coefficients so systems can be solved more quickly and consistently
What does the solution to a system of inequalities represent on a graph?
It represents all points that satisfy every inequality in the system
How can you tell whether to use an open or closed circle when graphing an inequality?
closed circle for ≥ or ≤; use an open circle for > or <
When graphing a system of two equations, the solution(s) of the system is/are the __________ of the graphs.
intersection
What does the determinant of a matrix tell you about the number of solutions?
A nonzero determinant means one unique solution; a zero determinant means no solution or infinitely many.
How can you tell if a point is in the feasible region?
Plug the point into each inequality; if it makes all of them true, it is inside the feasible region or graph and see if the point is in the region
What does the shaded region above a line mean when graphing a linear inequality?
all points with y-values greater than the line satisfy the inequality
What does it mean when a system has more than one solution when looking at the graph of two equations?
they intersect more than once
What does it mean for a matrix to have an inverse?
A matrix has an inverse only when it represents a system with one unique solution.
What does shading the region below a line mean in inequality graphs?
Shading below means the inequality describes all points with y-values less than the line.
What does it mean if a point lies in the solution region of an inequality?
It means the point makes the inequality true when substituted into the inequality
What is the main idea behind solving a system by substitution?
Solve one equation for a variable, then plug it into the other to reduce the system to a single equation
Why can’t you take the inverse of a matrix whose determinant is zero?
If the determinant is zero, the matrix cannot be inverted because it doesn’t correspond to a unique solution
Why do systems of inequalities often form polygons or bounded regions?
Because shading multiple regions produces an overlapping area where all conditions are true, often forming a polygon.
When solving inequalities, when do you flip the sign?
divide or multiply by a negative
How can you tell if a system is inconsistent without solving it?
Lines that are parallel (same slope, different intercepts) will never meet — meaning no solutions.
Why is writing a system as a matrix helpful for solving?
simplifies the system and allows solution methods such as matrix inversion
What does it mean when the feasible region is empty?
There is no set of points that satisfies all the inequalities at once