Inequalities
A. Linear Inequalities in 2 Variables
Closed half-plane: sign is > or < ; points on the boundary line are part of the solution set (solid line)
Open half-plane: < or > ; points on the boundary line are NOT part of the solution set (broken line)
Boundary line: divides the plane & describes the limit of the graph of an inequality
Boundary points: points on the boundary line
Solution region: part of the Cartesian plane where you will find the solution set
Half plane: how a solution region looks like
-x+y–4<0
-x + y – 4 = 0
-x + y = 4
(-4 , 0)
(0 , 4)
1
<
Equivalent equation
Standard form
x - intercept
y - intercept
Slope
Inequality symbol
How to shade
-x + y – 4 < 0
-x + y – 4 = 0
-x + y = 4
(-4 , 0)
(0, 4)
1
<
Inequality symbol
y < mx + b
y > mx + b
y < mx + b
Kind of boundary line
How to shade
Broken
Below the line
Direction of line
+
Slant to the right
/
Solid
Below
Broken
Above the line
Kind of slope
-0
Slant to the left
Horizontal
\
–
x+y>2
x+y=2
x+y=2
(2 , 0)
(0, 2)
-1
>
y > mx + b
Solid
above
undefined
Vertical
|
If y is negative, don’t forget to FLIP the inequality symbol.
Ex. -4x – 3y > 18  -3y > 4x + b
 -3y > 4x + 18
-3
-3
 y< 
4
x –6
3
B. Systems of Linear Inequality in 2 Variables
Solution for the system: set of all points that satisfy ALL the given inequalities in the system and is located
WITHIN the overlapping
regions of inequalities & the solid boundary line.

To check if a point is a solution for the system, substitute the values into ALL the inequalities in the system.
Ex. (5,4)
 y  2x  7

3x  2y  4

 y  2x  7
(4) 2(5)  7
4  10  7
14  7
3x  2y  4
3(5)  2(4)  4
15  8  4
7  4
since both are true, then (-5,4) is a solution of the system


Graphing System of Linear Equations in Two Variables
x  y  1
2x  y  7
Ex. 1 
first identify the x- and y-intercepts of each inequality.
Intercepts:

(-1,0) and (0,1)
7
( ,0) and (0, 7)
2
next, graph the lines and shade their respective solution regions

Since the inequalities use the > and < symbols,
the boundary lines are solid.
For x – y > -1, since y is negative, you flip the
symbol so it becomes <. Since it is LESS than,
you shade downwards. (blue part)
For 2x + y > 7, since y is positive, you retain the
symbol. For GREATER than, you shade
upwards. (red part)
The part where the red & blue regions overlap is
the solution region.
Ex. 2
y  4

x  6
21x  14 y  42


intercepts:y
(0,4)  horizontal line
(6,0)  vertical line
(2,0) ; (0, 3 5 )
7

For y > 4 and x < 6, use solid lines. For the third
inequality, since the symbol is < , use a broken line.
For y > 4, there is no x intercept and the slope is 0 that’s
why we use a horizontal line intercepting only the y axis.
Shade upwards. (blue)
For x < 6, there is no y intercept therefore the slope is
undefined. We use a vertical line intercepting only the x
axis and shade to the left. (red)
For 21x + 14y < 42, the symbol is LESS than, so you
shade downwards. (green)
The part where the red, blue and green regions overlap is
the solution region.
C. Solving word problems
Ex.
In basketball, a player scores 2 points for a basket and 1 point for a free throw. Suppose a player has scored no
more than 20 points in a game. How many baskets (b) and free throws (f) could the player have made?
I. Let f be the # of free throws
b be the # of baskets
2b  f  20

II. b  0
f  0

III.

find the intercepts...
2b  f  20
2b  20
2
b  10  (10, 0)
f  20  (0, 20)
The answer can be any of the points found in
the solution region. Don’t forget to check
your answers!

A.
Solution region
x and y axes are
supposed to be
BROKEN because
of > symbol.
Therefore, the
points on the axes
are NOT part of
the solution
region.