168 
… 5.6 Graphing Linear Inequalities
„‹Please graph the solution set for each inequality.
ƒˆˆˆ2x + 3y σ 6

ˆˆˆ x ³ y
ˆ 2x + 3y = 6ŠΔΔΕΔΔ
ˆˆˆ 0 ³ 2
ˆˆˆ 3 ³ 0
Gƒ [y <= -.666x + 2] 5,5,430,151,3,14
‚S
To find the equation of the boundry line to the solution set of the
inequality 2x + 3y σ 6, you should replace the 'σ' symbol with '=' to
give 2x + 3y = 6.

The graph of this equation is the boundry line of the solution set of
the given inequality.‹You should then graph this line by finding the
intercepts.
ˆˆˆˆŠ x ³ y
ˆˆˆˆŠ ΔΔΕΔΔ
ˆˆˆˆŠ 0 ³ 2
ˆˆˆˆŠ 3 ³ 0

The two intercepts (0,2) and (3,0) are plotted on the coordinate system
and a solid line is drawn through the two points.‹The line is drawn
solid whenever the inequality symbol is 'σ' or 'ς'.‹If the inequality
symbol is '>' or '<' the line drawn through the two points is dotted.
Since this is the boundry to the solution set, the solution set is either
to the left or right of the line.‹We can identify which side by trying
a point not on the line in the orginal inequality. A convenient point to
try is the origin, (0,0).
ˆˆˆˆ‹2x + 3y σ 6
#ˆˆˆˆ2ω0 + 3ω0 σ 6 ΔΔĝ‹0 σ 8 true

Since this is a true statement, we know the solution set is on the left
side of the line.‹We therefore shade in to the left side of the line.
If the statement had been false we would have shaded in to the right
side of the line.
Any ordered pair in the shaded region is a solution to 2x + 3y σ 6, and
any ordered pair not in the shaded region is not a solution.

G 1Š Graph the solution set of the inequality, 3x+ 4y σ 12.
A) y <= .75x + 3
B) y <= -.75x + 3
C) y <= -.75x - 3
D) y <= .75x - 3
3x + 4y σ 12
‰ 3x + 4y = 12ˆˆˆˆ‹Trying the origin (0,0)
ˆ‹x ³ yˆˆˆˆ‰ in the original in-
ˆ‹ΔΔΕΔĈˆˆˆ‰ equality gives 0 σ 12.
ˆ‹0 ³ 3ˆˆˆˆ‰ This is true so shade in
ˆ‹4 ³ 0ˆˆˆˆ‰ to the left of the line.
€ B
G 2Š Graph the solution set of the inequality, 4x - 2y > 8.
A) y > 2x - 4
B) y > -2x - 4
C) y > 2x + 4
D) y > -2x + 4
4x - 2y > 8
‰ 4x - 2y = 8ˆˆˆˆŠTrying the origin (0,0)
ˆ‹x ³ yˆˆˆˆ‰ in the original in-
ˆ‹ΔΔΕΔĈˆˆˆ‰ equality gives 0 > 8.
ˆ‹0 ³-4ˆˆˆˆ‰ This is false so shade
ˆ‹2 ³ 0ˆˆˆˆ‰ to the right of the line.
€ A
G 3‰ Graph the solution set of the inequality, x < 2.
A) y < 0x + 2
B) y < -999x - 2
C) y < -999x + 2
D) y < 0x - 2
‹x < 2
‹This is a verticalˆˆˆ‰ Trying the origin (0,0)
‹line with x-interceptˆˆˆŠin the original in-
‹'2'.‹x = 2ˆˆˆˆ‰ equality gives 0 < 2.
ˆˆˆˆˆˆ‰This is true so shade in
ˆˆˆˆˆˆ‰to the left of the line.
€ C
G 4‰Graph the solution set of the inequality, y ς -3.
A) y >= -999x - 3
B) y >= -999x + 3
C) y >= 0x + 3
D) y >= 0x - 3
 y ς -3
‹This is a horizontalˆˆˆ‰ Trying the origin (0,0)
‹line with y-ˆˆˆˆˆin the original in-
‹intercept '-3'.ˆˆˆˆŠ equality gives 0 ς -3.
ˆy = -3ˆˆˆˆˆ This is true so shade in
ˆˆˆˆˆˆˆabove the line.

€ A
G 5‰Graph the solution set of the inequality, 5x - 2y σ 10.
A) y <= -2.5x - 5
B) y >= 2.5x - 5
C) y >= -2.5x - 5
D) y <= .6x - 3
5x - 2y σ 10
‰ 5x - 2y = 10ˆˆˆˆ Trying the origin (0,0)
ˆ‹x ³ yˆˆˆˆ‰in the original in-
ˆ‹ΔΔΕΔĈˆˆˆ‰equality gives 0 σ 10.
ˆ‹0 ³-5ˆˆˆˆ‰This is true so shade in
ˆ‹2 ³ 0ˆˆˆˆ‰to the left of the line.

€ B
G 6‰Graph the solution set of the inequality, -3x + 5y > 15.
A) y > .6x + 3
B) y > -.6x + 3
C) y < -.6x - 3
D) y < -.6x + 3
-3x + 5y > 15
‰ -3x + 5y = 15ˆˆˆˆTrying the origin (0,0)
ˆŠx ³ yˆˆˆˆŠ in the original in-
ˆŠΔΔΕΔĈˆˆˆŠ equality gives 0 > 15.
ˆŠ0 ³ 3ˆˆˆˆŠ This is false, shade in
ˆ‹-5 ³ 0ˆˆˆˆŠ to the left of the line.

€ A
G 7‰Graph the solution set of the inequality, 3x + 4y ς 12
A) y >= .75x + 3
B) y >= .75x - 3
C) y >= -.75x + 3
D) y >= -.75x - 3
3x + 4y ς 12
‰ 3x + 4y = 12ˆˆˆˆ Trying the origin (0,0)
ˆ‹x ³ yˆˆˆˆ‰in the original in-
ˆ‹ΔΔΕΔĈˆˆˆ‰equality gives 0 ς 12.
ˆ‹0 ³ 3ˆˆˆˆ‰This is false, shade in
ˆ‹4 ³ 0ˆˆˆˆ‰to the right of the line.

€ C
G 8‰Graph the solution set of the inequality, x > -2.
A) y < -999x + 2
B) y < 0x + 2
C) y > -999x - 2
D) y > 0x - 2
 x > -2
‹This is a verticalˆˆˆ‰ Trying the origin (0,0)
‹line with x-ˆˆˆˆ‰in the original in-
‹intercept '-2'.ˆˆˆˆ‹equality gives 0 > -2.
ˆx = -3ˆˆˆˆ‰ This is true so shade in
ˆˆˆˆˆˆ‰to the right of the line.

€ C
G 9‰Graph the solution set of the inequality, y ς 1.
A) y <= -999x - 2
B) y <= 0x - 1
C) y >= 0x + 1
D) y >= -999x + 2
 y ς 1
‹This is a horizontalˆˆˆ‰ Trying the origin (0,0)
‹line with y-ˆˆˆˆˆin the original in-
‹intercept '1'.ˆˆˆˆ‰equality gives 0 ς 1.
ˆy = 1ˆˆˆˆˆ‹This is false, shade in
ˆˆˆˆˆˆˆabove this line.

€ C
G 10‰ Graph the solution set of the inequality, y < 2x.
A) y < 2x + 0
B) y < -2x + 0
C) y < .5x + 0
D) y > -.5x + 0
 y < 2x
ˆy = 2xˆˆˆˆˆTrying another point like
ˆx ³ yˆˆˆˆˆ (-1,1) in the original
ˆΔΔΕΔĈˆˆˆˆ inequality gives 1 < -2.
ˆ0 ³ 0ˆˆˆˆˆ This is false, shade in
ˆ1 ³ 2ˆˆˆˆˆ to the right of the line.

€ A