We are given a compound inequality. If we solve each inequality separately, we will find two solution sets. The union of those sets, the set containing values from one set or from the other, is the solution to the compound inequality.
First Inequality
Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must reverse the inequality sign.
The second inequality is satisfied for all values of c less than or equal to 3. In this case c=3 is included in the solution set.
Comparing Solution Sets
Now we must compare our solution sets. From the first inequality, c >6, we know that the solution set consists of all numbers to the right of 6 on the number line, without including 6 itself. We will graph this by using an open circle at 6.
From the second inequality c ≤ 3, we know the solution set includes all values to the left of 3, including c=3. We will graph this using a closed circle at 3.
The union of these solution sets is two intervals, c >6 or c ≤ 3.
