To solve the inequality [tex]\(-6 - x \leq 7\)[/tex], let's go through the steps:
1. Isolate the variable [tex]\(x\)[/tex]:
[tex]\[
-6 - x \leq 7
\][/tex]
Add 6 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
-x \leq 7 + 6
\][/tex]
Simplify the right-hand side:
[tex]\[
-x \leq 13
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
To remove the negative sign in front of [tex]\(x\)[/tex], multiply both sides of the inequality by -1. Remember, multiplying or dividing an inequality by a negative number reverses the inequality sign:
[tex]\[
x \geq -13
\][/tex]
3. Interpreting the result:
The inequality [tex]\(x \geq -13\)[/tex] means any [tex]\(x\)[/tex] value that is greater than or equal to -13 will satisfy the inequality.
Now, let's test the given values to see which ones satisfy the inequality:
- I. [tex]\(-13\)[/tex]:
[tex]\[
-13 \geq -13 \text{ (True)}
\][/tex]
- II. [tex]\(-14\)[/tex]:
[tex]\[
-14 \geq -13 \text{ (False)}
\][/tex]
- III. [tex]\(-6\)[/tex]:
[tex]\[
-6 \geq -13 \text{ (True)}
\][/tex]
Based on the evaluations:
- [tex]\(-13\)[/tex] satisfies the inequality.
- [tex]\(-14\)[/tex] does not satisfy the inequality.
- [tex]\(-6\)[/tex] satisfies the inequality.
Therefore, the values that satisfy the inequality [tex]\(-6 - x \leq 7\)[/tex] are [tex]\(-13\)[/tex] and [tex]\(-6\)[/tex]. Thus, the correct answer is:
I and III
