Absolute value linear inequalities are inequalities containing one or more absolute expressions of linear terms.
A simple example of absolute value linear inequalities would be The universal way to solve these is to divide the absolute value expression into two cases: when the term inside is positive, or negative. The case when the expression is exactly zero can be included in either one of the two cases. Let's see how it works through an example.
When the sign is e.g. the result will be
When the sign is e.g. the result will be and
The red portions are our answers.
We know that when and when Thus we divide it into those two cases.
(i) When the term is equivalent to Hence we have However, since we have assumed that the values of that satisfy the given inequality under this condition is
(ii) When the term is equivalent to Hence we have However, since we have assumed that this time we have
After solving each case, we find the union of the solution sets of each case, which is the answer. Therefore our answer is
If there are more than one absolute value expression (e.g. ), then we do the same thing for each expression. When we have absolute value expressions, we will have to divide it into cases (unless an expression is the multiple of another one, like and ). Then the union of the solution sets of each case is the answer.
Inequalities with a single absolute value expression can be approached in a slightly easier way. Note that is equivalent to or and is equivalent to Using this principle, the example above can be solved as thus:
Using the principle mentioned above, we have
Now let's try solving some more example problems.
If then which implies If then which implies Hence, the solution is or
Observe that the given inequality is equivalent to
Then, if we have If we have Therefore, the answer is or
How many integers satisfy
We can rewrite the inequality as follows to obtain:
Thus, three integers satisfy the inequality, so the answer is
How many integer solutions does the following inequality have:
If we have
If we have
If we have
Hence, from the above three cases we have which implies that five integers satisfy the given inequality. So our answer is
Let the solutions to the inequality be or where and are constants. What is
Observe that implies or which is equivalent to
Then since we are given or we have the following two equations:
which implies and Therefore,