Question

Determine if the point (3, 8) is a solution to the simultaneous inequalities: y > x + 5 and y ≤ 6x - 3. Why?

Answer 1

To determine if the point (3, 8) is a solution to the simultaneous inequalities \( y > x + 5 \) and \( y \leq 6x - 3 \), we need to check both inequalities one at a time.

1. **Check the first inequality \( y > x + 5 \)**:
We substitute \( x = 3 \) and \( y = 8 \) into the inequality:
\[
8 > 3 + 5
\]
Simplifying the right side:
\[
8 > 8
\]
This inequality is **not true** because 8 is not greater than 8.

2. **Check the second inequality \( y \leq 6x - 3 \)**:
We again substitute \( x = 3 \) and \( y = 8 \) into this inequality:
\[
8 \leq 6(3) - 3
\]
Simplifying the right side:
\[
8 \leq 18 - 3
\]
\[
8 \leq 15
\]
This inequality is **true**.

Since the point (3, 8) does not satisfy the first inequality but satisfies the second inequality, the point (3, 8) is **not a solution** to the simultaneous inequalities.

In conclusion:
- The point (3, 8) fails the first inequality \( y > x + 5 \), even though it satisfies the second inequality \( y \leq 6x - 3 \).

Thus, the final answer is that (3, 8) is NOT a solution to the simultaneous inequalities.