Solve for xx and graph the solution on the number line below.
minus, x, minus, 3, is greater than or equal to, minus, 2, or, minus, x, minus, 3, is less than or equal to, minus, 9
−x−3≥−2or
−x−3≤−9
Let's first solve each inequality:
For the first inequality:
−x - 3 ≥ −2
Add 3 to both sides to isolate the term with x:
−x ≥ −2 + 3
−x ≥ 1
Since the inequality has a negative sign in front of x, to solve for x, we need to divide both sides by -1 (remember that dividing by a negative number will flip the inequality sign):
x ≤ −1
For the second inequality:
−x - 3 ≤ −9
Add 3 to both sides:
−x ≤ −9 + 3
−x ≤ −6
Divide by -1 (and flip the inequality sign):
x ≥ 6
Now we have two inequalities:
x ≤ −1 (solution for the first inequality)
x ≥ 6 (solution for the second inequality)
The word "or" between these two inequalities means that we are looking for a union of the solutions, not the intersection.
The solution on the number line is two separate ranges:
x is to the left of -1 (including -1 itself) and to the right of 6 (including 6 itself).
To represent these two solutions on a number line:
```
←--(----[-1--------------------------------6]----)-->
```
The parentheses, "(" and ")", indicate open endpoints, which are not part of the solution,
but since we have "greater than or equal to" and "less than or equal to," we use closed brackets, "[", "]", to show that -1 and 6 are included in the solution set:
Note: If I could provide you with an actual number line graphic, you would see -1 and 6 marked on the line with the "rays" heading towards their respective directions with brackets indicating they are part of the solution.