Let's determine whether the test point [tex]\((-4, 7)\)[/tex] satisfies the linear inequality [tex]\(x + y \leq 3\)[/tex].
To do this, we need to substitute [tex]\(x = -4\)[/tex] and [tex]\(y = 7\)[/tex] into the given inequality and check if the resulting statement is true.
### Step-by-Step Solution:
1. Identify the coordinates of the test point:
- [tex]\(x = -4\)[/tex]
- [tex]\(y = 7\)[/tex]
2. Substitute the values into the inequality:
- Original inequality: [tex]\(x + y \leq 3\)[/tex]
- Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = 7\)[/tex]:
[tex]\[
-4 + 7 \leq 3
\][/tex]
3. Simplify the left side of the inequality:
- Calculate [tex]\(-4 + 7\)[/tex]:
[tex]\[
-4 + 7 = 3
\][/tex]
4. Rewrite the inequality with the simplified left side:
[tex]\[
3 \leq 3
\][/tex]
5. Determine if the statement is true:
- The statement [tex]\(3 \leq 3\)[/tex] is true since 3 is equal to 3.
Since the inequality holds true when substituting the given values, the test point [tex]\((-4, 7)\)[/tex] satisfies the inequality [tex]\(x + y \leq 3\)[/tex].
### Conclusion:
B. The test point [tex]\((-4, 7)\)[/tex] is a solution to the inequality because substituting [tex]\(-4\)[/tex] for [tex]\(x\)[/tex] and [tex]\(7\)[/tex] for [tex]\(y\)[/tex] makes the inequality a true statement.
