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, x, minus, 4
y<−x−4
y, is greater than, start fraction, 5, divided by, 4, end fraction, x, plus, 1
y>
4
5
​
x+1
x
y
Answer
Multiple Choice Answers
left parenthesis, 1, comma, minus, 8, right parenthesis(1,−8)
left parenthesis, 6, comma, 5, right parenthesis(6,5)
left parenthesis, minus, 6, comma, minus, 3, right parenthesis(−6,−3)
left parenthesis, minus, 2, comma, 3, right parenthesis(−2,3)

To determine which point satisfies both inequalities, let's analyze each inequality separately and then check the given points:

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

First, we will substitute each point into both inequalities to see if they satisfy the conditions.

### Point (1, -8)
1. \( y < -x - 4 \)
   \[ -8 < -1 - 4 \]
   \[ -8 < -5 \] (True)
2. \( y > \frac{5}{4}x + 1 \)
   \[ -8 > \frac{5}{4}(1) + 1 \]
   \[ -8 > 1.25 + 1 \]
   \[ -8 > 2.25 \] (False)

### Point (6, 5)
1. \( y < -x - 4 \)
   \[ 5 < -6 - 4 \]
   \[ 5 < -10 \] (False)
2. \( y > \frac{5}{4}x + 1 \)
   \[ 5 > \frac{5}{4}(6) + 1 \]
   \[ 5 > 7.5 + 1 \]
   \[ 5 > 8.5 \] (False)

### Point (-6, -3)
1. \( y < -x - 4 \)
   \[ -3 < 6 - 4 \]
   \[ -3 < 2 \] (True)
2. \( y > \frac{5}{4}x + 1 \)
   \[ -3 > \frac{5}{4}(-6) + 1 \]
   \[ -3 > -7.5 + 1 \]
   \[ -3 > -6.5 \] (True)

### Point (-2, 3)
1. \( y < -x - 4 \)
   \[ 3 < 2 - 4 \]
   \[ 3 < -2 \] (False)
2. \( y > \frac{5}{4}x + 1 \)
   \[ 3 > \frac{5}{4}(-2) + 1 \]
   \[ 3 > -2.5 + 1 \]
   \[ 3 > -1.5 \] (True)

Based on this analysis, the point that satisfies both inequalities is:

\[ \boxed{(-6, -3)} \]

Would you like further details or have any questions? Here are 5 additional questions you might find useful:

1. How do you solve systems of linear equations?
2. What are the properties of linear inequalities?
3. Can you explain how to graph inequalities on a coordinate plane?
4. How do you determine the feasible region for a system of inequalities?
5. What is the difference between strict and non-strict inequalities?

Tip: When graphing inequalities, always start by graphing the boundary line, then use a test point to determine where to shade.

Learn how to identify points satisfying two linear inequalities y < -x - 4 and y > 5/4x + 1. Explore step-by-step solutions and understand graphing on a coordinate plane. Suitable for students in Grades 9-12.

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