91Ó°ÊÓ

Solving inequalities involves finding all possible values of a variable that satisfy the given inequality. For absolute value inequalities, such as \(|-5c - 4| > 8\), we generally follow these steps:

First, we isolate the absolute value expression. In the example, we do this by subtracting 2 from both sides of the inequality to get \[8 < |-5c - 4|\].

Next, we split the inequality into two separate inequalities based on the definition of absolute value. For \(|x| > a\), this means we have \[x > a\] or \[x < -a\]. Applying this to our isolated term, we get the inequalities \[-5c - 4 > 8\] and \[-5c - 4 < -8\].

Then, we solve the individual inequalities. Make sure to pay close attention especially when multiplying or dividing by a negative number, as this will change the direction of the inequality.

Once we solve the inequalities, we need to express the solution in interval notation. Interval notation provides a compact way of writing ranges of values.

For the inequality \[-5c - 4 > 8\], solving it gives us \[c < -2.4\]. For \[-5c - 4 < -8\], solving it gives us \[c > 0.8\].

The solutions \[-2.4\] and \[0.8\] result in two separate intervals. We write the result as the union of the two intervals: