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 half, x, minus, 1

*y*<−21​*x*−1

y, is less than, start fraction, 4, divided by, 5, end fraction, x, minus, 2

*y*<54​*x*−2

12345678910-1-2-3-4-5-6-7-8-9-1012345678910-1-2-3-4-5-6-7-8-9-10xy

Answer

Multiple Choice Answers

left parenthesis, 7, comma, 8, right parenthesis(7,8)

left parenthesis, 0, comma, minus, 7, right parenthesis(0,−7)

left parenthesis, 9, comma, 1, right parenthesis(9,1)

left parenthesis, minus, 3, comma, minus, 3, right parenthesis(−3,−3)

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

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

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

### Point (7, 8)
1. \( y < -\frac{1}{2}x - 1 \)
   \[ 8 < -\frac{1}{2}(7) - 1 \]
   \[ 8 < -3.5 - 1 \]
   \[ 8 < -4.5 \] (False)
2. \( y < \frac{4}{5}x - 2 \)
   \[ 8 < \frac{4}{5}(7) - 2 \]
   \[ 8 < 5.6 - 2 \]
   \[ 8 < 3.6 \] (False)

### Point (0, -7)
1. \( y < -\frac{1}{2}x - 1 \)
   \[ -7 < -\frac{1}{2}(0) - 1 \]
   \[ -7 < -1 \] (True)
2. \( y < \frac{4}{5}x - 2 \)
   \[ -7 < \frac{4}{5}(0) - 2 \]
   \[ -7 < -2 \] (True)

### Point (9, 1)
1. \( y < -\frac{1}{2}x - 1 \)
   \[ 1 < -\frac{1}{2}(9) - 1 \]
   \[ 1 < -4.5 - 1 \]
   \[ 1 < -5.5 \] (False)
2. \( y < \frac{4}{5}x - 2 \)
   \[ 1 < \frac{4}{5}(9) - 2 \]
   \[ 1 < 7.2 - 2 \]
   \[ 1 < 5.2 \] (True)

### Point (-3, -3)
1. \( y < -\frac{1}{2}x - 1 \)
   \[ -3 < -\frac{1}{2}(-3) - 1 \]
   \[ -3 < 1.5 - 1 \]
   \[ -3 < 0.5 \] (True)
2. \( y < \frac{4}{5}x - 2 \)
   \[ -3 < \frac{4}{5}(-3) - 2 \]
   \[ -3 < -2.4 - 2 \]
   \[ -3 < -4.4 \] (False)

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

\[ \boxed{(0, -7)} \]

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

1. How do you graph linear inequalities?
2. What are some strategies for solving systems of inequalities?
3. How can you check if a point is a solution to an inequality?
4. What is the significance of the slope in a linear inequality?
5. How do you interpret the y-intercept in the context of an inequality?

Tip: When solving inequalities, always double-check your calculations by substituting points back into the original inequalities.

Explore how to identify points that satisfy multiple inequalities through detailed analysis. This math problem involves analyzing linear inequalities and graphing to find valid solutions. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation