Solving Compound Inequalities

If we solve each inequality separately, we will find two solution sets. Since the compound inequality includes the word "or," the union of those sets is the solution to the compound inequality.

First Inequality

We solve inequalities the same way we would solve equations. We just need to keep in mind that multiplying or dividing by a negative number reverses the sign of the inequality. Here, we'll add $8$ to both sides so the inequality sign doesn't change. All possible values of $x$ that are less than or equal to $12$ will satisfy the inequality. Note that $12$ is included in the solution set.

Second Inequality

In this case, we can isolate $x$ by subtracting $3$ from both sides of the inequality. The second inequality is satisfied for all values of $x$ greater than $3.$ In this case, $3$ is not included in the solution set.

Comparing Solution Sets

Now we must compare our solution sets. From the first inequality, $x\le 12,$ we know that the solution set consists of all numbers to the left of $12$ on the number line, including $12$ itself. We will graph the endpoint with a closed circle at $12.$

From the second inequality, $x>3,$ we know the solution set includes all values to the right of $3,$ without including $3.$ Thus, we'll graph the endpoint with an open circle at $3.$

The union of these solution sets is all real numbers.

This means that any real number will solve the first, second, or both inequalities.