Question

Select all the solutions that satisfy the system of inequalities:

There are two

y>x2-4

y≤4x-2

a
(0,-2)

b
(2,0)

c
(0,-5)

d
(0,-4)

Answer 1

To determine which points satisfy the system of inequalities, we can evaluate each point against the two inequalities:

1. \( y > x^2 - 4 \)
2. \( y \leq 4x - 2 \)

Let's evaluate each point:

### Point A: (0, -2)
1. For \( y > x^2 - 4 \):
\[
-2 > 0^2 - 4 \implies -2 > -4 \quad \text{(True)}
\]
2. For \( y \leq 4x - 2 \):
\[
-2 \leq 4(0) - 2 \implies -2 \leq -2 \quad \text{(True)}
\]

**Point A satisfies both inequalities.**

### Point B: (2, 0)
1. For \( y > x^2 - 4 \):
\[
0 > 2^2 - 4 \implies 0 > 4 - 4 \implies 0 > 0 \quad \text{(False)}
\]
2. For \( y \leq 4x - 2 \):
\[
0 \leq 4(2) - 2 \implies 0 \leq 8 - 2 \implies 0 \leq 6 \quad \text{(True)}
\]

**Point B does not satisfy the first inequality.**

### Point C: (0, -5)
1. For \( y > x^2 - 4 \):
\[
-5 > 0^2 - 4 \implies -5 > -4 \quad \text{(False)}
\]
2. For \( y \leq 4x - 2 \):
\[
-5 \leq 4(0) - 2 \implies -5 \leq -2 \quad \text{(True)}
\]

**Point C does not satisfy the first inequality.**

### Point D: (0, -4)
1. For \( y > x^2 - 4 \):
\[
-4 > 0^2 - 4 \implies -4 > -4 \quad \text{(False)}
\]
2. For \( y \leq 4x - 2 \):
\[
-4 \leq 4(0) - 2 \implies -4 \leq -2 \quad \text{(True)}
\]

**Point D does not satisfy the first inequality.**

### Summary of Results:
- Point A: (0, -2): **Satisfies both inequalities**
- Point B: (2, 0): **Does not satisfy the first inequality**
- Point C: (0, -5): **Does not satisfy the first inequality**
- Point D: (0, -4): **Does not satisfy the first inequality**

The only solution that satisfies the system of inequalities is:

**a (0, -2)**