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, start fraction, 3, divided by, 4, end fraction, x, plus, 5
y<− 
4
3
​
 x+5
y, is less than, start fraction, 5, divided by, 2, end fraction, x, minus, 2
y< 
2
5
​
 x−2
x
y
Answer
Multiple Choice Answers
left parenthesis, 9, comma, 3, right parenthesis(9,3)
left parenthesis, minus, 1, comma, 8, right parenthesis(−1,8)
left parenthesis, minus, 3, comma, minus, 7, right parenthesis(−3,−7)
left parenthesis, 10, comma, minus, 7, right parenthesis(10,−7)

To determine which point satisfies both inequalities, we need to check each point in the multiple-choice answers to see if it makes both inequalities true.

The given inequalities are:

1. \( y < -\frac{3}{4}x + 5 \)
2. \( y < \frac{5}{2}x - 2 \)

We will substitute the \(x\) and \(y\) values from each point into both inequalities and check the validity.

**1. Point \( (9, 3) \)**

- **First Inequality:**
  \[
  3 < -\frac{3}{4}(9) + 5
  \]
  \[
  3 < -\frac{27}{4} + 5
  \]
  \[
  3 < -\frac{27}{4} + \frac{20}{4}
  \]
  \[
  3 < -\frac{7}{4}
  \]
  This is **false**.

- **Second Inequality:**
  \[
  3 < \frac{5}{2}(9) - 2
  \]
  \[
  3 < \frac{45}{2} - 2
  \]
  \[
  3 < 22.5 - 2
  \]
  \[
  3 < 20.5
  \]
  This is **true**.

Since the first inequality is false, point \( (9, 3) \) does not satisfy both inequalities.

**2. Point \( (-1, 8) \)**

- **First Inequality:**
  \[
  8 < -\frac{3}{4}(-1) + 5
  \]
  \[
  8 < \frac{3}{4} + 5
  \]
  \[
  8 < \frac{3}{4} + \frac{20}{4}
  \]
  \[
  8 < \frac{23}{4}
  \]
  \[
  8 < 5.75
  \]
  This is **false**.

- **Second Inequality:**
  \[
  8 < \frac{5}{2}(-1) - 2
  \]
  \[
  8 < -\frac{5}{2} - 2
  \]
  \[
  8 < -\frac{5}{2} - \frac{4}{2}
  \]
  \[
  8 < -\frac{9}{2}
  \]
  \[
  8 < -4.5
  \]
  This is **false**.

Since both inequalities are false, point \( (-1, 8) \) does not satisfy both inequalities.

**3. Point \( (-3, -7) \)**

- **First Inequality:**
  \[
  -7 < -\frac{3}{4}(-3) + 5
  \]
  \[
  -7 < \frac{9}{4} + 5
  \]
  \[
  -7 < \frac{9}{4} + \frac{20}{4}
  \]
  \[
  -7 < \frac{29}{4}
  \]
  \[
  -7 < 7.25
  \]
  This is **true**.

- **Second Inequality:**
  \[
  -7 < \frac{5}{2}(-3) - 2
  \]
  \[
  -7 < -\frac{15}{2} - 2
  \]
  \[
  -7 < -\frac{15}{2} - \frac{4}{2}
  \]
  \[
  -7 < -\frac{19}{2}
  \]
  \[
  -7 < -9.5
  \]
  This is **false**.

Since the second inequality is false, point \( (-3, -7) \) does not satisfy both inequalities.

**4. Point \( (10, -7) \)**

- **First Inequality:**
  \[
  -7 < -\frac{3}{4}(10) + 5
  \]
  \[
  -7 < -\frac{30}{4} + 5
  \]
  \[
  -7 < -\frac{30}{4} + \frac{20}{4}
  \]
  \[
  -7 < -\frac{10}{4}
  \]
  \[
  -7 < -2.5
  \]
  This is **true**.

- **Second Inequality:**
  \[
  -7 < \frac{5}{2}(10) - 2
  \]
  \[
  -7 < \frac{50}{2} - 2
  \]
  \[
  -7 < 25 - 2
  \]
  \[
  -7 < 23
  \]
  This is **true**.

Since both inequalities are true, point \( (10, -7) \) satisfies both inequalities.

**Conclusion:** The point that satisfies both inequalities is **\( (10, -7) \)**.

---

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

1. How do you graph inequalities on a coordinate plane?
2. What are the differences between linear inequalities and linear equations?
3. How do you find the intersection of two inequalities graphically?
4. What is the significance of the slope in a linear inequality?
5. How do you determine the solution set for a system of inequalities?
6. What are the methods to solve a system of linear inequalities algebraically?
7. Can you explain how to use test points to determine the solution region for inequalities?
8. How do inequalities relate to real-world scenarios?

**Tip:** When graphing inequalities, always remember to use a dashed line for a strict inequality ("<" or ">") and a solid line for an inclusive inequality ("≤" or "≥").

Explore how to find points that satisfy two linear inequalities through a detailed problem-solving approach. This problem involves graphing inequalities and testing points to determine valid solutions. Suitable for Grades 9-12 students studying linear inequalities in coordinate geometry.

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