|x+9| is greater than or equal to 0. Solve and graph the solution set on a number line.
The is 10/9
|x+9| ≥ 0
x+9 ≥ 0 or -x-9 ≥ 0
x ≥ -9 or x ≤ -9
so x is any real number
We could have seen that from the start
|x+9| ≥ 0 by the very definition of the absolute value.
|anything| ≥ 0
So would the set be (-9,�‡)?
To solve the inequality |x+9| ≥ 0, we need to consider two cases.
Case 1: x+9 ≥ 0
In this case, the absolute value of (x+9) equals (x+9) since it is already non-negative and does not change sign. So the inequality becomes x+9 ≥ 0.
Solving this inequality, we subtract 9 from both sides:
x+9-9 ≥ 0 - 9
x ≥ -9
So for Case 1, the solution is x ≥ -9.
Case 2: -(x+9) ≥ 0
In this case, the absolute value of (x+9) equals -(x+9) since it is negative and we need to change its sign. So the inequality becomes -(x+9) ≥ 0.
Solving this inequality, we multiply both sides by -1 (which reverses the inequality sign):
-(x+9) * -1 ≤ 0 * -1
x+9 ≤ 0
Now we subtract 9 from both sides:
x+9-9 ≤ 0-9
x ≤ -9
So for Case 2, the solution is x ≤ -9.
Now we can combine the solutions from both cases:
x ≥ -9 and x ≤ -9.
On the number line:
──────────|──────────
-9 - - - - - - - - -
The overlapping region represents the solution set, which is x ≤ -9 and x ≥ -9.
Therefore, the solution set for |x+9| ≥ 0 is x ≤ -9 or x ≥ -9.