Question

graphing linear inequality systems
aurora garcia
graphing linear inequality systems (a.rei.12)
systems of linear inequalities
system of linear inequalities
$y = -3x + 6$
$y = x + 2$
one solution:
$(1, 3)$
$y > -3x + 6$
$0 > -3(0) + 6$
$0 > 0 + 6$
$0 > 6$
false
$y \leq x + 2$
$0 \leq 0 + 2$
$0 \leq 2$
true
how many solutions?
infinite solutions
$y = 2x + 6$
no solution
$y = -x + 8$
$2y = -2x + 16$
infinite
solutions
$y > 2x + 6$
$0 > 2(0) + 6$
$0 > 0 + 6$
$0 > 6$
false
$y \leq 2x - 1$
$0 \leq 2(0) - 1$
$0 \leq 0 - 1$
$0 \leq -1$
false
how many solutions?
no solution
multiple-choice question
what types of solution sets are possible with a system of inequalities?
check all that apply
no solution points
infinitely many solution points
1 solution point

Explanation:

• For a system of linear inequalities, the solution set can be:
• No solution points: When the regions defined by the inequalities do not overlap (e.g., \( y > 2x + 6 \) and \( y \leq 2x - 1 \) as shown in the graph, their regions have no intersection).
• Infinitely many solution points: When the overlapping region of the inequalities is a non - empty area (e.g., the system with \( y > - 3x + 6 \) and \( y\leq x + 2 \) has an infinite number of points in the overlapping region).
• A system of linear inequalities cannot have exactly 1 solution point because the solution to a linear inequality is a region (a set of points), not a single point (unlike a system of linear equations which can have a single point as a solution).

Answer:

• no solution points
• infinitely many solution points