Graphing Inequalities
Graph: y < 3x−1
Should the line be solid or dashed?
Dashed
Solve
y=x+3
y=2x+5
Answer x = -2, y = 1
f(x) =
x+2 x < 1
4 x ≥ 1
f (0) = 0+2=2
f (2) = 4
System:
y <= x + 2
y >= 0
Q: Which quadrant(s) contain the solution?
A: Quadrants I and II
x+y = 5
x-y = 1
Add the equations: 2x=6 -> x=3, then y = 2
y≥ -x + 3
Which side do you shade?
Above the line
y=3x-1
2x+y=11
x=2.4, y=6.2
f(x) =
2x - 1 if x < 2
x^2 if x >= 2
Evaluate f (1) and f (3)
f (1) = 1, f (3) = 9
System:
y > x
y <= 3
Q: Is the point (1,2) a solution?
A: Yes, 2 > 1 and 2 <= 3
2x+3y=12
4x-3y=6
y=2, x = 3
Which inequality matches a solid line and shading below?
A) y<mx+b
B) y≤mx+b
C) y>mx+b
D) y≥mx+b
B) y≤mx+b
y=1/2x + 1
3x-2y=4
f(x) =
x + 5 if x < 3
2x - 1 if x >= 3
Is f(x) continuous at x = 3?
No, Left = 8, Right = 5 → Not continuous
System:
y >= 2x - 1
y < -x + 4
Q: Does the point (1,2) satisfy the system?
Yes, 2 >= 1 ✅and 2 < 3
3x+2y=14
2x-3y=4
x= 50/13
y = 16/13
Test point (1,2) for y > x-1. If false, what do you do?
Shade the opposite side
A student substitutes y = 2x – 3 into 4x + y = 9. Is this correct for substitution? Why?
Yes substitution means replacing y with 2x-3. Then solve 4x + (2x-3) = 9 -> 6x-3 = 9 -> x = 2
f(x) =
-x + 2 if x < 0
x^2 - 1 if x >= 0
Find all x where f(x) = 3
x = -1, 2
System:
y > -x + 1
y <= 2x + 2
Q: Pick a point NOT on a line that satisfies the system.
A: Example: (1,2) → 2 > 0 and 2 <= 4
3x+2y=7
6x-4y=15
No solution
Graph the inequality
y > −2x+3
Is it solid or dashed? Which side of the line should be shaded? Would the point (2,0) be in the solution region?
Dashed, Above the line, (2,0) would work.
y= 3/2x - 1
4x - 2y = 10
x = 8
y = 11
f(x) =
3x - 1 if x <= 2
-x^2 + 6 if x > 2
Find f(2) + f(3)
f(2) = 5, f(3) = -3 → Sum = 2
System:
y >= x - 3
y < -x + 5
Q: Describe the solution region in words.
(gl w this one)
A: Triangular region where points are above y = x - 3 but below y = -x + 5 -> roughly between x = -1 and x = 4
5x-2y=7
10x-4y=14
Infinitely many solutions