inequalities

1. D. J. Jayamanne
NSBM 1
Inequalities

2. • An inequality is like an equation, but instead of an equal sign (=)
it has one of these signs:
• < : less than
• ≤ : less than or equal to
• > : greater than
• ≥ : greater than or equal to
• An inequality is a mathematical sentence that uses one of the
inequality symbols to state the relationship between two
quantities.
Inequality?
NSBM 2

3. Examples
NSBM 3
Symbol Words Example
> greater than x + 3 > 2
< less than 7x < 28
≥ greater than or equal to 5 ≥ x - 1
≤ less than or equal to 2y + 1 ≤ 7

4. • When we graph an inequality on a number line we use open and
closed circles to represent the number.
Graphing Inequalities
NSBM 4
<
<
Plot a closed circle
≤ ≥
Plot an open circle

5. 0 5 10 15
-20 -15 -10 -5
-25 20 25
•The numbers 4, 3, 2, 1, 0, -1, -2, -3, etc. are less than 5.
• There are also numbers in between the integers, like 2.5, 1/2, -7.9, etc.
• The number 5 is not less than 5. Therefore the inequality will not be
true when x=5.
Graph x<5 on a number line.
5

6. • Numbers greater than -2 are to the right of -2 on the number line.
• The number -2 would also be a correct answer, because of the
phrase, “or equal to”.
0 5 10 15
-20 -15 -10 -5
-25 20 25
-2
6
Graph x ≥ -2 on a number line.

7. • Follow the same rules and steps that we used to solve an
equation.
• Always undo addition or subtraction first, then multiplication.
• Remember whatever is done to one side of the inequality must
be done to the other side. The goal is to get the variable by itself.
7
Solving an Inequality

8. • When you multiply or divide each side of an inequality by a
negative number, you must reverse the inequality symbol to
maintain a true statement.
8
THE TRAP…..

9. Multiply each side by the same negative number and REVERSE the
inequality symbol.
4

 x Multiply by (-1).
4


x
See the switch
9
Solving by multiplication of a negative #

10. Divide each side by the same negative number and reverse the
inequality symbol.
2
6
2
2




x
3


x
6
2 
 x
10
Solving by dividing by a negative #

11. Solve the following inequalities and graph the answer
1. w+5 < 8
2. 8 + r ≥ -2
3. -2x > 2
4. 2h + 8 ≤ 24
5. x + 3 > -4
6. 6d > 24
7. 2x - 8 < 14
8. -2c - 4 ≤ 2
11

12. 12
Inequality notation vs. Interval notation
0 4 6 8
2
-7 -6 -5 -4 -3 -2 -1 1 5 7
3
1


x
Inequality notation for
the graph shown above.
)
,
1
[ 

Interval notation for the
graph shown above.

13. ]
4
,
2
(
Let's try another one.
2
-7 -6 -5 -4 -3 -2 -1 1 5 7
3
0 4 6 8
Rounded bracket means
cannot equal -2
Squared bracket means
can equal 4
The brackets used in the interval notation above are the same ones
used when you graph this.
This means everything between –2 and 4 but not including -2
13

14. Compound Inequalities
• Let's consider a "double inequality" (having two inequality signs).
• x is in-between the two numbers. This is an "and" inequality which
means both parts must be true. It says that x is greater than –2
and x is less than or equal to 3.
14
6
4
0 8
2
-7 -6 -5 -4 -3 -2 -1 1 5 7
3 2
-7 -6 -5 -4 -3 -2 -1 1 5 7
3
0 4 6 8
3
2 

 x
( ]

15. Compound Inequalities
• Let's consider a "double inequality" (having two inequality signs).
• Instead of "and", these are "or" problems. One part or the other
part must be true (but not necessarily both). Either x is less than –
2 or x is greater than or equal to 3. In this case both parts cannot
be true at the same time since a number can't be less than –2 and
also greater than 3.
15
6
4
0 8
2
-7 -6 -5 -4 -3 -2 -1 1 5 7
3 2
-7 -6 -5 -4 -3 -2 -1 1 5 7
3
0 4 6 8
3
or
2 

 x
x
) [

16. 3
or
2 

 x
x
There are two intervals to list when you list in
interval notation.
2
-7 -6 -5 -4 -3 -2 -1 1 5 7
3
0 4 6 8
) [
)
2
,
( 
 )
,
3
[ 
When the solution consists of more than one interval,
we join them with a union sign.

16

17. Represent the following inequalities on a number line
1. x < 8 or x > 20
2. -1< x < 3
3. -1 < x < 4
4. -1 < x < 4
5. x + 3 > -4 or x < -12
17

18. • A linear inequality in two variables can be written in any one of
these forms:
 Ax + By < C
 Ax + By > C
 Ax + By ≤ C
 Ax + By ≥ C
• An ordered pair (x, y) is a solution of the linear inequality if the
inequality is TRUE when x and y are substituted into the
inequality.
18
Linear Inequalities

19. Example 1
• Which ordered pair is a solution of
5x - 2y ≤ 6?
A. (0, -3)
B. (5, 5)
C. (1, -2)
D. (3, 3)
19

20. • The graph of a linear inequality is the set of all points in a
coordinate plane that represent solutions of the inequality.
• We represent the boundary line of the inequality by drawing
the function represented in the inequality. The boundary line
will be a
• Solid line when ≤ and ≥ are used.
• Dashed line when < and > are used.
20
Graphing Linear Inequalities

21. • Our graph will be shaded on one side of the boundary line to show
where the solutions of the inequality are located.
• Test a point NOT on the boundary line to determine which side of the
line includes the solutions. (The origin is always an easy point to test,
but make sure your line does not pass through the origin)
 If your test point is a solution (makes a TRUE statement),
shade THAT side of the boundary line.
 If your test points is NOT a solution (makes a FALSE
statement), shade the opposite side of the boundary line.
21
Graphing Linear Inequalities

22. • Graph the inequality x ≤ 4 in a coordinate plane.
• Decide whether to
use a solid or
dashed line.
• Use (0, 0) as a
test point.
• Shade where the
solutions will be.
22
Example 1
y
x
5
5
-5
-5

23. • Graph 3x - 4y > 12 in a coordinate plane.
• Sketch the boundary line of the graph.
 Find the x- and
y-intercepts and
plot them.
• Solid or dashed
line?
• Use (0, 0) as a
test point.
• Shade where the
solutions are.
y
x
5
5
-5
-5
23
Example 2

24. • Graph y < 2/5x in a coordinate plane.
• Sketch the boundary line of the graph.
 Find the x- and y-intercept and plot them.
 Both are the origin!
• Use the line’s slope
to graph another point.
• Solid or dashed
line?
• Use a test point
OTHER than the
origin.
• Shade where the
solutions are.
y
x
5
5
-5
-5
24
Example 3

25. Plot the following inequalities in the coordinate plane
1. x ≥ -2
2. y < 3
3. x > y
4. y > 4x − 5
5. y > 4x − 5
6. y >-2x+3
7. y >-2x+3
25