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Below are two inequalities and the graphs of their lines without the shading. By imagining where the shading should be, identify which point would satisfy BOTH inequalities.
y, is less than, minus, one third, x, plus, 6
y<− 
3
1
​
 x+6
y, is less than, 6, x, minus, 1
y<6x−1
x
y
Answer
Multiple Choice Answers
left parenthesis, minus, 3, comma, 9, right parenthesis(−3,9)
left parenthesis, minus, 1, comma, minus, 1, right parenthesis(−1,−1)
left parenthesis, 0, comma, minus, 9, right parenthesis(0,−9)
left parenthesis, 10, comma, 10, right parenthesis(10,10)

To determine which point satisfies both inequalities, we need to check each point in the multiple-choice answers to see if they satisfy both inequalities:

1. \( y < -\frac{1}{3}x + 6 \)
2. \( y < 6x - 1 \)

Let's substitute each point into the inequalities:

### 1. Point \((-3, 9)\)

- **First Inequality**:
  \[
  9 < -\frac{1}{3}(-3) + 6
  \]
  \[
  9 < 1 + 6
  \]
  \[
  9 < 7
  \]
  This is **false**.

- **Second Inequality**:
  \[
  9 < 6(-3) - 1
  \]
  \[
  9 < -18 - 1
  \]
  \[
  9 < -19
  \]
  This is **false**.

Since both inequalities are false, point \((-3, 9)\) does not satisfy both inequalities.

### 2. Point \((-1, -1)\)

- **First Inequality**:
  \[
  -1 < -\frac{1}{3}(-1) + 6
  \]
  \[
  -1 < \frac{1}{3} + 6
  \]
  \[
  -1 < \frac{1}{3} + \frac{18}{3}
  \]
  \[
  -1 < \frac{19}{3}
  \]
  \[
  -1 < 6.33
  \]
  This is **true**.

- **Second Inequality**:
  \[
  -1 < 6(-1) - 1
  \]
  \[
  -1 < -6 - 1
  \]
  \[
  -1 < -7
  \]
  This is **false**.

Since the second inequality is false, point \((-1, -1)\) does not satisfy both inequalities.

### 3. Point \((0, -9)\)

- **First Inequality**:
  \[
  -9 < -\frac{1}{3}(0) + 6
  \]
  \[
  -9 < 0 + 6
  \]
  \[
  -9 < 6
  \]
  This is **true**.

- **Second Inequality**:
  \[
  -9 < 6(0) - 1
  \]
  \[
  -9 < 0 - 1
  \]
  \[
  -9 < -1
  \]
  This is **true**.

Since both inequalities are true, point \((0, -9)\) satisfies both inequalities.

### 4. Point \((10, 10)\)

- **First Inequality**:
  \[
  10 < -\frac{1}{3}(10) + 6
  \]
  \[
  10 < -\frac{10}{3} + 6
  \]
  \[
  10 < -\frac{10}{3} + \frac{18}{3}
  \]
  \[
  10 < \frac{8}{3}
  \]
  \[
  10 < 2.67
  \]
  This is **false**.

- **Second Inequality**:
  \[
  10 < 6(10) - 1
  \]
  \[
  10 < 60 - 1
  \]
  \[
  10 < 59
  \]
  This is **true**.

Since the first inequality is false, point \((10, 10)\) does not satisfy both inequalities.

**Conclusion:** The point that satisfies both inequalities is **\((0, -9)\)**.

---

Do you have any questions or need further clarification? Here are some related questions you might explore:

1. How do you determine the solution set for a system of linear inequalities?
2. What does the intersection of two inequalities represent on a graph?
3. How can you use a test point to verify the solution region for an inequality?
4. What are the differences between solving linear equations and inequalities?
5. How does the slope and y-intercept of a line affect the inequality solutions?
6. Can you describe the steps to graph a linear inequality?
7. How do you determine if a system of inequalities has no solution?
8. What is the significance of boundary lines in inequalities?

**Tip:** Remember, always substitute the points carefully into inequalities and simplify the equations step by step to avoid errors.

Explore how to identify points that satisfy two linear inequalities through an example problem. Learn to solve systems of inequalities and graphing techniques. Suitable for Grades 9-12 students.

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