Notice that the given solution is a compound inequality. It consists of two absolute value inequalities.
Compound:& |x| ≥ 5 and |x| ≤ 6
Inequality I:& |x| ≥ 5
Inequality II:& |x| ≤ 6
Therefore, we should first graph the absolute value inequalities separately and then combine their graphs. Let's start with Inequality I.
Inequality I
The absolute value inequalities with the signs > and ≥ can be written as a compound inequality with the word or.
Absolute Value:& |x| ≥ 5
Compound:& x ≥ 5 or x ≤ -5
The compound inequality says that the values greater than or equal to 5 orless than or equal to -5 are solutions to this absolute value inequality. Therefore, its graph will be the union of the graphs of the inequalities which form it. Notice that since the inequalities are non-strict, we will show the endpoints with closed points.
Inequality II
If the absolute value inequalities are formed by the signs < or ≤, they can be written as compound inequalities with the word and.
Absolute Value:& |x| ≤ 6
Compound:& x ≥ -6 and x ≤ 6
In this case, the solutions to the inequality are the values greater than or equal to -6 andless than or equal to 6. The graph of the compound inequality will be the intersection of the graphs of the inequalities that form it. Because the inequalities are non-strict, endpoints will be closed.
Combining the Solution Sets
Since the given solution is combined with the word and, the graph of it will be the intersection of the graphs of the absolute value inequalities that form it.
