Since the word between the inequalities is or, we are looking for the union of the solution sets to the individual inequalities.
Solution Set: {a | a>5} Graph:
Practice makes perfect
To solve the compound inequality, we have to solve each of the inequalities separately. Since the word between the individual inequalities is or, the solution set for the compound inequality is the union of the individual solutions.
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 reverse the inequality sign.
The above tells us that all values greater than or equal to 6 will satisfy this inequality.
Note that the point on 6 is closed because it is included in the solution set.
Second inequality
In the second inequality the variable is already isolated so we can graph it immediately. This inequality tells us that all values greater than 5 will be in the solution set.
Note that the point on 5 is open because it is not included in the solution set.
Compound inequality
The solution to the compound inequality is the union of the solution sets. Notice that the solution set to the first inequality lies within the solution set of the second inequality. Therefore, the combined solution set can be represented by the second inequality only.
First Solution Set:& a≥ 6
Second Solution Set:& a > 5
Combined Solution Set:& a>5
The above can be written using set notation.
{a | a>5}
Finally, we will graph the solution set to the compound inequality.
