The solution set of 9−x2≥∣x+3∣ is....
Explanation
The absolute value has the definition
∣x+3∣={x+3;for x≥−3−(x+3);for x<−3
Solve the inequality
For x≥−3
9−x2≥∣x+3∣
9−x2≥x+3
−x2−x+6≥0
(−x+2)(x+3)≥0
x=2∨x=−3
Create the number line
Number Line for x≥−3
Interval [−3,2] indicates both endpoints are included in the solution set.
−
+
−
−3
2
From the condition x≥−3 and the number line region above, the solution set is {−3≤x≤2}.
For x<−3
9−x2≥∣x+3∣
9−x2≥−(x+3)
−x2+x+12≥0
−(x−4)(x+3)≥0
x=4∨x=−3
The number line
Number Line for x<−3
Interval [−3,4] indicates both endpoints are included in the solution set.
−
+
−
−3
4
From the condition x<−3 and the second number line region above, the solution set is an empty set.
Therefore the combined solution is
HP={−3≤x≤2}∪{}
HP={−3≤x≤2}