Question

Is (–4, 10) a solution to this system of inequalities?

y ≥ –1/4x+ 9
y < –1/2x+ 9

Answer 1

To determine if the point \((-4, 10)\) is a solution to the given system of inequalities, we need to check whether it satisfies each inequality.

1. **First Inequality:**
\[
y \geq -\frac{1}{4}x + 9
\]
Substitute \(x = -4\) and \(y = 10\):
\[
10 \geq -\frac{1}{4}(-4) + 9
\]
Calculate the right side:
\[
-\frac{1}{4}(-4) = 1 \quad \Rightarrow \quad 1 + 9 = 10
\]
So the inequality becomes:
\[
10 \geq 10
\]
This is true.

2. **Second Inequality:**
\[
y < -\frac{1}{2}x + 9
\]
Substitute \(x = -4\) and \(y = 10\):
\[
10 < -\frac{1}{2}(-4) + 9
\]
Calculate the right side:
\[
-\frac{1}{2}(-4) = 2 \quad \Rightarrow \quad 2 + 9 = 11
\]
So the inequality becomes:
\[
10 < 11
\]
This is true.

Since the point \((-4, 10)\) satisfies both inequalities, it is indeed a solution to the system.