SECTION 2.8 Solving Linear Inequalities^151
OBJECTIVES
Solving Linear Inequalities
2.8
1 Graph intervals on
a number line.
2 Use the addition
property of
inequality.
3 Use the
multiplication
property of
inequality.
4 Solve linear
inequalities by
using both
properties of
inequality.
5 Solve applied
problems by using
inequalities.
6 Solve linear
inequalities with
three parts.
An inequalityis an algebraic expression related by
“is less than,” “is less than or equal to,”
7 “is greater than,” or Ú “is greater than or equal to.”
6 ...
Linear Inequality in One VariableA linear inequality in one variablecan be written in the form
,,,or
where A, B, and Crepresent real numbers, and AZ0.
AxB<C AxB◊C AxB>C AxB»C,
Some examples of linear inequalities in one variable follow.
and Linear inequalities
We solve a linear inequality by finding all real number solutions of it. For example,
the solution set
z- 2 k+ 5 ... 10
3
4
x+ 56 2, Ú5,
⎩⎪⎨⎪⎧ ⎩⎨⎧ ⎩⎪⎨⎪⎧ ⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧
–4
–3 –2 –1 041 2 3
2 is included.FIGURE 18Graph of the interval 1 - q, 2 4The set of numbers less than or equal to 2 is an example of an intervalon the
number line. We can write this interval using interval notation.
Interval notationThe negative infinitysymbol does not indicate a number, but shows that the in-
terval includes allreal numbers less than 2. Again, the square bracket indicates that
2 is part of the solution.
ˆ
1 - q, 2 4
The set of all x such that xis less than or equal to 25 x|x... 26
includesall real numbersthat are less than or equal to 2, not just theintegersless
than or equal to 2.
OBJECTIVE 1 Graph intervals on a number line.Graphing is a good way to
show the solution set of an inequality. To graph all real numbers belonging to the set
, we place a square bracket at 2 on a number line and draw an arrow
extending from the bracket to the left (since all numbers less than2 are also part of
the graph). See FIGURE 18.
5 x|x... 26
Set-builder notation
(Section 1.4)