We were asked to solve a compound inequality. Let's start by splitting it into separate inequalities.
Compound Inequality:&& 9≤ 6-b& < 12
First Inequality:&& 9≤ 6-b&
Second Inequality:&& 6-b& < 12
Notice that compound inequalities written in this way are equivalent to compound inequalities that involve the word and.
9≤ 6-b and 6-b < 12Let's solve the inequalities separately.
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 flip the inequality sign.
The above tells us that by all values greater than - 6 will satisfy the second inequality.
Note that the point on - 6 is open because it is not included in the solution set.
Combining Solution Sets
The solution set to the compound inequality is the intersection of the solution sets.
First Solution Set: b& ≤ - 3
Second Solution Set: - 6< b&
Intersecting Solution Set: -6 < b& ≤ - 3
Finally, we will graph the solution set to the compound inequality on a number line.
