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. $$ y\frac{5}{6} x-4 $$

Question

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. $$ y<-\frac{5}{6} x+2 $$ $$ y>\frac{5}{6} x-4 $$

UpStudy AI · Accepted Answer

To determine which point satisfies both inequalities, we need to analyze each inequality and understand the regions they represent on a graph.

1. **First Inequality:**
   $$
   y < -\frac{5}{6} x + 2
   $$
   This inequality represents the area below the line $$ y = -\frac{5}{6} x + 2 $$. The line has a slope of $$-\frac{5}{6}$$ and a y-intercept of $$2$$.

2. **Second Inequality:**
   $$
   y > \frac{5}{6} x - 4
   $$
   This inequality represents the area above the line $$ y = \frac{5}{6} x - 4 $$. The line has a slope of $$\frac{5}{6}$$ and a y-intercept of $$-4$$.

### Steps to Identify the Region:

- **Graph the lines:**
  - For the first line $$ y = -\frac{5}{6} x + 2 $$:
    - When $$ x = 0 $$, $$ y = 2 $$ (point $$ (0, 2) $$).
    - When $$ y = 0 $$, $$ 0 = -\frac{5}{6} x + 2 $$ leads to $$ x = \frac{12}{5} $$ (point $$ \left(\frac{12}{5}, 0\right) $$).
  
  - For the second line $$ y = \frac{5}{6} x - 4 $$:
    - When $$ x = 0 $$, $$ y = -4 $$ (point $$ (0, -4) $$).
    - When $$ y = 0 $$, $$ 0 = \frac{5}{6} x - 4 $$ leads to $$ x = \frac{24}{5} $$ (point $$ \left(\frac{24}{5}, 0\right) $$).

### Finding the Intersection:
To find the point that satisfies both inequalities, we can find the intersection of the two lines by setting them equal to each other:

$$
-\frac{5}{6} x + 2 = \frac{5}{6} x - 4
$$

Now, let's solve for $$ x $$:

1. Add $$\frac{5}{6} x$$ to both sides:
   $$
   2 = \frac{5}{6} x + \frac{5}{6} x - 4
   $$
   $$
   2 = \frac{10}{6} x - 4
   $$

2. Add $$4$$ to both sides:
   $$
   6 = \frac{10}{6} x
   $$

3. Multiply both sides by $$\frac{6}{10}$$:
   $$
   x = \frac{36}{10} = \frac{18}{5}
   $$

Now substitute $$ x = \frac{18}{5} $$ back into one of the original equations to find $$ y $$:

Using the first equation:
$$
y = -\frac{5}{6} \left(\frac{18}{5}\right) + 2 = -\frac{18}{6} + 2 = -3 + 2 = -1
$$

Thus, the intersection point is $$ \left(\frac{18}{5}, -1\right) $$.

### Conclusion:
The point $$ \left(\frac{18}{5}, -1\right) $$ satisfies both inequalities, as it lies in the region below the first line and above the second line.
The point $$ \left(\frac{18}{5}, -1\right) $$ satisfies both inequalities.

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. \[ y<-\frac{5}{6} x+2 \] \( y>\frac{5}{6} x-4 \)

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