Lesson 1-6 – Compound Inequalities
Compound Inequality – consists of two inequalities joined by the word and (conjunction) or the word or
(disjunction).
To solve an “and” compound inequality:
Method 1 -- Solve each part of the inequality separately, then graph the solution set
Method 2 -- Solve both parts at the same time making sure that you perform each operation to both
sides of the individual inequalities then graph the solution set.
To graph the solution set on a number line:
Graph the solution set for each inequality and find their intersection (area of overlap)
The graph of an “and” compound inequality is called the intersection of the solution set of the two
inequalities.
To solve an “or” compound inequality:
Solve each inequality separately and then graph the solution set
or
To graph the solution set on a number line:
Graph each solution set separately and then combine on one number line
The graph of an “or” compound inequality is the union of the solution set of the two inequalities.
Solving Absolute Value Inequalities
Reminder: The absolute value of a number is its distance from 0 on the number line. It is always nonnegative.
To solve an absolute value inequality (<) or (≤)
Using the definition of absolute value, graph the solution set on a number line
Use the information on the graph to write out the solution set
Ex:
*This means that all the values of d are less than 3 units from 0 on the number line.
Graph on a number line
All the solutions are between
and
and are less than
Now use this information to write out the solution set
and
or
In Conclusion:
Since
We can say that if:
Is the same as
then
then
and
units from .
Solving an Absolute Value Inequality (>) or (≥)
Using the definition of absolute value, graph the solution set on a number line
Use the information on the graph to write out the solution set
Ex:
*This means that all the values for
are at least
units or more from
on the number line.
Graph on a number line
The distance between
and
on the number line is greater than or equal to units
Now use this information to write out the solution set.
or
In conclusion:
Since
Is the same as
We can say that if:
or
then
or
then
or
Because the absolute value of a number is never negative:
The solution of an inequality like
is all real numbers
The solution of an inequality like
is the empty set
Solving a multistep absolute value inequality and graph the solution set.