We were asked to solve a compound inequality. Let's start by splitting it into separate inequalities.
Compound Inequality: 6< x&+ 5 ≤ 11
First Inequality: 6< x&+5
Second Inequality: x&+5 ≤ 11
Notice that compound inequalities written in this way are equivalent to compound inequalities that involve the word and.
6< x+5 and x+5 ≤ 11
Let'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, we must flip the inequality sign.
We found that all values less than or equal to 6 will satisfy the inequality.
Note that the point on 6 is closed because it is included in the solution set.
Compound Inequality
The solution set of the compound inequality is the intersection of the solution sets.
First Solution Set: 1< x&
Second Solution Set: x&≤ 6
Intersecting Solution Set: 1< x& ≤ 6
Finally, we will graph the solution set to the compound inequality on a number line.
