Answer:
"Solving'' an inequality means finding all of its solutions. A "solution'' of an inequality is a number which when substituted for the variable makes the inequality a true statement. When we substitute 8 for x, the inequality becomes 8-2 > 5. Thus, x=8 is a solution of the inequality.
Step-by-step explanation:
Example 1:
Consider the inequality
displaymath173
The basic strategy for inequalities and equations is the same: isolate x on one side, and put the "other stuff" on the other side. Following this strategy, let's move +5 to the right side. We accomplish this by subtracting 5 on both sides (Rule 1) to obtain
displaymath174
after simplification we obtain
displaymath175
Once we divide by +2 on both sides (Rule 3a), we have succeeded in isolating x on the left:
displaymath176
or simplified,
displaymath177
All real numbers less than 1 solve the inequality. We say that the "set of solutions'' of the inequality consists of all real numbers less than 1. In interval notation, the set of solutions is the interval tex2html_wrap_inline187 .
Example 2:
Find all solutions of the inequality
displaymath189
Let's start by moving the ``5'' to the right side by subtracting 5 on both sides (Rule 1):
displaymath190
or simplified,
displaymath191
How do we get rid of the ``-'' sign in front of x? Just multiply by (-1) on both sides (Rule 3b), changing " tex2html_wrap_inline201 " to " tex2html_wrap_inline203 " along the way:
displaymath192
or simplified
displaymath193
All real numbers greater than or equal to -1 satisfy the inequality. The set of solutions of the inequality is the interval tex2html_wrap_inline205 .
Example 3:
Solve the inequality
displaymath207
Let us simplify first:
displaymath208
There is more than one route to proceed; let's take this one: subtract 2x on both sides (Rule 1).
displaymath209
and simplify:
displaymath210
Next, subtract 9 on both sides (Rule 1):
displaymath211
simplify to obtain
displaymath212
Then, divide by 4 (Rule 3a):
displaymath213
and simplify again:
displaymath214
It looks nicer, if we switch sides (Rule 2).
displaymath215
In interval notation, the set of solutions looks like this: tex2html_wrap_inline227 .