Question

Determine the relationship between the point (1, –5) and the given system of inequalities.

y ≤ 3x + 2

y > –2x – 3

Explain your answer both algebraically and graphically.

Answer 1

Algebraic check:
- For y ≤ 3x + 2: substitute (1, −5): −5 ≤ 3(1) + 2 = 5. This is true.
- For y > −2x − 3: substitute (1, −5): −5 > −2(1) − 3 = −5. This is false (−5 is equal to −5, not greater).

Since the point fails the second (strict) inequality, it does not satisfy the system.

Graphical explanation:
- Graph y = 3x + 2 as a solid line (because ≤ includes the line) and shade the region at or below it.
- Graph y = −2x − 3 as a dashed line (because > excludes the line) and shade the region above it.
- The point (1, −5) lies exactly on the line y = −2x − 3, so it is on the dashed boundary and is not in the shaded region for y > −2x − 3. Although it is in the shaded region for the first inequality, being on the excluded boundary of the second means it is not a solution to the system.