Question

In Exercises 3–8, tell whether the ordered pair is
a solution of the system of linear inequalities.
(See Example 1.)
3. (1, 2);
y ≥ 5x − 6
y ≤ 3x + 1
y > −x + 2


6.(−2, 5);
y ≤ x + 3
y ≥ −x
3x − y > −6

Answer 1

To determine whether the ordered pairs \((1, 2)\) and \((-2, 5)\) are solutions of the given systems of linear inequalities, we will substitute the \(x\) and \(y\) values of each ordered pair into each inequality and check if they satisfy all inequalities.

### Problem 3: Evaluation of \((1, 2)\)
**Inequalities:**
1. \(y \geq 5x - 6\)
2. \(y \leq 3x + 1\)
3. \(y > -x + 2\)

**Substituting \( (1, 2) \):**
1. **First Inequality:**
\(2 \geq 5(1) - 6\)
\(2 \geq 5 - 6\)
\(2 \geq -1\) (True)

2. **Second Inequality:**
\(2 \leq 3(1) + 1\)
\(2 \leq 3 + 1\)
\(2 \leq 4\) (True)

3. **Third Inequality:**
\(2 > -1 + 2\)
\(2 > 1\) (True)

Since the point \((1, 2)\) satisfies all three inequalities, it **is** a solution to the system.

### Problem 6: Evaluation of \((-2, 5)\)
**Inequalities:**
1. \(y \leq x + 3\)
2. \(y \geq -x\)
3. \(3x - y > -6\)

**Substituting \( (-2, 5) \):**
1. **First Inequality:**
\(5 \leq -2 + 3\)
\(5 \leq 1\) (False)

2. **Second Inequality:**
\(5 \geq -(-2)\)
\(5 \geq 2\) (True)

3. **Third Inequality:**
\(3(-2) - 5 > -6\)
\(-6 - 5 > -6\)
\(-11 > -6\) (False)

Since the point \((-2, 5)\) does not satisfy the first and third inequalities, it **is not** a solution to the system.

### Summary:
- \((1, 2)\) is a solution to the system of inequalities in problem 3.
- \((-2, 5)\) is not a solution to the system of inequalities in problem 6.