Lamar is buying vitamins for his dog. The daily dose for the vitamins is based on the dog’s weight. Lamar’s dog weighs 32 pounds. Since 32 is greater than 25, but less than or equal to 50, he will give his dog 2 tablets.
Another way to write this information
is to use an inequality. Let w represent the weight that requires 2
tablets.
These two inequalities form a compound inequality.
The compound inequality w > 25 and w ≤ 50 can be written
without using the word and.
Note that in each, both inequality symbols are facing the
same direction.
1.
Write x ≥ 2 and x < 7 as a
compound inequality without using and.
Alternative Solutions:
x ≥ 2 and x < 7 can be written
as 2 ≤ x < 7 or as 7 > x ≥ 2.
A compound inequality using and is
true if and only if both inequalities are true. Thus, the graph of a
compound inequality using and is the intersection of the graphs of the two
inequalities.
Consider the inequality –2 < x < 3. It can be written using and:
x > –2 and x < 3. To graph, follow the steps below.
The solution, shown by the graph of
the intersection, is {x│–2 < x < 3}.
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2. Refer to the application at the beginning of the lesson. Graph the
solution of 25 < w < 50.
Alternative Solutions:
Rewrite
the compound inequality using and. 25 < w < 50 is the same
as w > 25 and w ≤ 50.
The
solution is {w│25 < w ≤ 50}.
Often, you must solve a compound inequality before graphing
it.
Example
3.
Solve 4 < x + 3 ≤ 12. Graph the
solution.
Alternative Solutions:
The
solution is {x│1 < x ≤ 9}.
The
graph of the solution is shown at the right.
Another type of compound inequality uses the word or.
This type of inequality is true if one or more of the inequalities is true. The
graph of a compound inequality using or is the union of the graphs of
the two inequalities.
4.
Graph the solution of x > 0 or x ≤ –1.
Alternative Solutions:
Sometimes you must solve compound
inequalities containing the word or before you are able to graph the solution.
Example
5.
Solve 3x ≥ 15 or –2x < 4. Graph the solution.
Alternative Solutions:
Now
graph the solution.
The last
graph shows the solution {x│x > –2}.
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