College Algebra 9th txtbk

60 CHAPTER 1 EQUATIONS AND INEQUALITIES
if they have the same solution set. Just as with equations, we perform operations on inequalities
that produce simpler equivalent inequalities, and continue the process until an inequality
is reached whose solution is obvious. The properties of inequalities given in Theorem 1
can be used to produce equivalent inequalities.
Z THEOREM 1 Inequality Properties
An equivalent inequality will result and the sense (or direction) will remain the
same if each side of the original inequality
• Has the same real number added to or subtracted from it
• Is multiplied or divided by the same positive number
An equivalent inequality will result and the sense (or direction) will reverse if
each side of the original inequality
• Is multiplied or divided by the same negative number
Note: Multiplication by 0 and division by 0 are not permitted.
Theorem 1 tells us that we can perform essentially the same operations on inequalities
that we perform on equations, with the exception that the sense (or direction) of the inequality
reverses if we multiply or divide both sides by a negative number: Otherwise the sense
of the inequality does not change.
Now let’s see how the inequality properties are used to solve linear inequalities. Examples
3, 4, and 5 will illustrate the process.
EXAMPLE 3 Solving a Linear Inequality
Solve and graph: 2(2x 3) 10 6 6(x 2)
SOLUTION
2(2x 3) 10 6 6(x 2)
4x 6 10 6 6x 12
4x 4 6 6x 12
4x 4 4 6 6x 12 4
Multiply out parentheses.
Combine like terms.
Add 4 to both sides.
4x 6 6x 8
4x 6x 6 6x 8 6x
2x 6 8
2x
2 7 8
2
Subtract 6x from both sides.
Divide both sides by 2.
Note that direction reverses
because 2 is negative.
x 7 4
or
(4, )
Solution set
( x
2 3 4 5 6 7 8 9
Graph of solution set
MATCHED PROBLEM 3
Solve and graph: 3(x 1) 5(x 2) 5