To make solving a little bit easier, we can separate the compound inequality into two cases.
Compound Inequality: 19 ≥&3z+1≥-5
First Inequality: 19 ≥&3z+1
Second Inequality: &3z+1≥-5
Notice that compound inequalities written in this way are equivalent to compound inequalities that involve the word and.
19 ≥3z+1 and 3z+1≥-5Let'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.
This above tells us that - 2 is less than or equal to all values that satisfy the inequality.
Note that the point on - 2 is closed because it is included in the solution set.
Compound Inequality
The solution set to the compound inequality is the intersection of the solution sets.
First Solution Set: z&≤ 6
Second Solution Set: -2 ≤ z&
Intersecting Solution Set: - 2≤ z& ≤ 6
Finally, we will graph the solution set to the compound inequality on a number line.
