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Linear inequalities
SLIDE 1
The relation , which is read “less than”, is an order relation.
? Order the following set by means of
? Which of these statements is true, and which is false?
(a)
(b)
(c)
(d)
SLIDE 2
? Order the following set by means of
Solution
? Which of these statements is true, and which is false?
Solution
(a)
(b)
(c)
(d)
SLIDE 3
Fundamental rule for solving equations in linear inequalities
If you multiply both sides by , then reverse the sign of the inequality.
Otherwise, equations in linear inequalities behave just like linear equations (equations with ).
Example
Solve for x
Solution
SLIDE 4
Solve
? ?
? ?
SLIDE 5
?
?
?
?
SLIDE 6
? What happens when we substitute into ?
? Is true when ?
? The following argument contains an algebraic error
How do we know that a mistake has been made with the algebra, and what is the mistake?
SLIDE 7
? When we have a 0 in the denominator of , so is undefined for .
? When .
Since , is false when .
? In the argument,
we know we have made a mistake because when the statement is false, and . The mistake is at the line marked **. At that line we multiply by , but if , then is negative and we should have changed the sign of the inequality to . But we cannot do this either, because we do not know at this stage whether x is positive or negative – so we cannot solve rational inequalities by this method.
SLIDE 8
Equations with rational inequalities
We cannot solve directly by converting it to an equation involving a linear inequality.
We will introduce the solution to rational inequalities at a higher level.
SLIDE 9
A linear function is a function of the form , where a and b are rational numbers.
A rational function is a function of the form
Exercise
Classify the following as linear or rational functions
? ?
? ?
SLIDE 10
?
?
?
?
SLIDE 11
Number puzzles with inequalities
If and which of the following is true? (More than one answer is possible.)
A B
C D
E F
SLIDE 12
Thus
A B
C D
E F
SLIDE 13
If and where a is an integer, which of the following is a solution set of all possible values of a?
SLIDE 14
SLIDE 15
The null set
If there is no solution to a problem, the solution set is the null set.
It is a set (collection) that is empty, that is, has no members.
The null set is represented by the symbol .
Example
Find the solution set to the inequalities .
Solution
There is no x such that . That is, is impossible. Hence, the solution set is .
SLIDE 16
State the solution set for each of the following, where x is an integer.
? ?
? ?
SLIDE 17
?
?
?
?
SLIDE 18
Given that , state whether the following are true or false.
?
?
?
?
SLIDE 19
This makes a and b negative numbers, and c a positive number
statement example
? true
? true
? true
? true
SLIDE 20
Given that , state whether the following are necessarily true or false.
?
?
?
?
SLIDE 21
all three numbers, a, b and c, are positive.
statement examples
? true
? true
? true
? true
SLIDE 22
Which of the following ordered pairs satisfies both the inequalities above?
A
B
C
D
E
SLIDE 23
We test each point in both equations
A ? ?
B ? ?
C ? ?
D ? ?
E ? ?
The answer is E
SLIDE 24
Eliminate x from these equations to find an inequality for y alone.
SLIDE 25
Eliminate x from these equations to find an inequality for y alone.
Solution
If then
Then
SLIDE 26
A jointed stick comprises three pieces of lengths a, b and c, where c is the longest length, and . It is to be folded at the joints to make a triangle.
Find a relation between the lengths a, b and c such that it will be impossible to make a triangle.
SLIDE 27
A jointed stick comprises three pieces of lengths a, b and c, where c is the longest length. It is to be folded at the joints to make a triangle.
Solution
This will not be possible if the sum of the lengths a and b are less than c.
If then we can make a line only, so for any triangle we have , where c is the longest side
SLIDE 28
The triangle inequality
For any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
SLIDE 29
The lengths of two sides of a triangle are and , where . The length of the third side is y. Find the range of possible values of y.
SLIDE 30
The lengths of two sides of a triangle are and , where . The length of the third side is y. Find the range of possible values of y.
Solution
Since , then both and , as we would expect, since they are lengths. But .
There are two possibilities, either is the longest side of the triangle, or the third side with length y is the longest side.
Case 1 is the longest side. Then by the triangle inequality and .
Case 2 y is the longest side. Then and .
Combining the two inequalities, and , we obtain
.
SLIDE 31
State whether these sets of three numbers could be the lengths of the sides of a triangle. If not, then explain why.
? 5, 12, 13 ? 3, 4, 6
? 5, 8, 13 ? 41, 59, 17
SLIDE 32
? 5, 12, 13 Can be a triangle.
In fact, it is a Pythagorean triple
? 3, 4, 6
The longest side is smaller than the sum of the other sides, so this is possible
? 5, 8, 13
This cannot be made into a triangle. It can only make a line.
? 41, 59, 17
This cannot make a triangle. The sum of the shorter lengths is smaller than the longest length.
SLIDE 33
Graphs of linear inequalities
An inequality of the form defines a region in the x,y-plane. To represent this region
? Recall that an exact inequality of the form is represented in the x,y-plane by a solid line, where as an inexact inequality of the form or is represented by a dashed line.
? The line divides the x,y-plane into two regions. This line can be sketched or plotted in the usual way. In , a is the gradient and b is the y-intercept.
? In one of these regions the inequality is true, and in the other it is false. Test a point to find whether that point lies in the region or not. The origin, , is usually a good test, as substituting into the inequality is simple.
? The region where the inequality holds is shown. For example, it may be shaded. Because the choice of shading is arbitrary, the region where the inequality is true should be labelled.
SLIDE 34
Example
Mark on the xy-plane the region where
Solution
The graph of has gradient 2 and y-intercept . We use a dashed line to indicate as the inequality is inexact. Substituting the coordinates of the origin, we get , which is true. Therefore, the origin lies in the region. We shade the area on that side of the line to indicate the region where is true (or “holds”), and mark the region R.
SLIDE 35
Mark on the xy-plane the region where
SLIDE 36
SLIDE 37
Mark on the xy-plane the region where both and hold.
SLIDE 38
SLIDE 39
Mark on the xy-plane the region where both and hold.
SLIDE 40
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