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Solving Compound Inequalities

Solving Compound Inequalities 1.8 - Solution

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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 to both sides so the inequality sign doesn't change.
All possible values of that are less than or equal to will satisfy the inequality. Note that is included in the solution set.

Second Inequality

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

Comparing Solution Sets

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

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

The union of these solution sets is all real numbers.

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