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

Accepted Answer

To solve this, we need to find a point that satisfies both inequalities $ y < -\frac{2}{5}x + 6 $ and $ y < 5x - 6 $. Let's assume we test a point, say $(2, 0)$:

### Step 1: Test the first inequality $ y < -\frac{2}{5}x + 6 $
Substitute $ x = 2 $ and $ y = 0 $:
$ 0 < -\frac{2}{5}(2) + 6 $
$ 0 < -\frac{4}{5} + 6 $
$ 0 < \frac{26}{5} $ (which is true, since $ \frac{26}{5} = 5.2 $)

### Step 2: Test the second inequality $ y < 5x - 6 $
Substitute $ x = 2 $ and $ y = 0 $:
$ 0 < 5(2) - 6 $
$ 0 < 10 - 6 $
$ 0 < 4 $ (which is also true)

Let's check another point, say $(3, 5)$:

### Step 1: First inequality
$ 5 < -\frac{2}{5}(3) + 6 $
$ 5 < -\frac{6}{5} + 6 $
$ 5 < \frac{24}{5} $ (which is false, since $ \frac{24}{5} = 4.8 $)

Now, let's check $(2, 0)$ again. Wait, maybe a better approach is to find the intersection of the lines and then check the region. The lines are $ y = -\frac{2}{5}x + 6 $ and $ y = 5x - 6 $. Let's find their intersection:

Set $ -\frac{2}{5}x + 6 = 5x - 6 $
Multiply both sides by 5: $ -2x + 30 = 25x - 30 $
$ 30 + 30 = 25x + 2x $
$ 60 = 27x $
$ x = \frac{60}{27} = \frac{20}{9} \approx 2.22 $
Then $ y = 5(\frac{20}{9}) - 6 = \frac{100}{9} - \frac{54}{9} = \frac{46}{9} \approx 5.11 $

Now, the region for $ y < -\frac{2}{5}x + 6 $ is below the first line, and $ y < 5x - 6 $ is below the second line. Let's check a point in the overlapping region. Let's take $ (3, 4) $:

### Step 1: First inequality
$ 4 < -\frac{2}{5}(3) + 6 $
$ 4 < -\frac{6}{5} + 6 $
$ 4 < \frac{24}{5} $ (false, since $ \frac{24}{5} = 4.8 $, but 4 < 4.8 is true? Wait, 4 is less than 4.8, so that's true. Wait, my mistake earlier. $ 4 < 4.8 $ is true.

### Step 2: Second inequality
$ 4 < 5(3) - 6 $
$ 4 < 15 - 6 $
$ 4 < 9 $ (true)

Wait, maybe I made a mistake with $(3, 5)$. Let's recheck $(3, 5)$ in the first inequality: $ 5 < -\frac{6}{5} + 6 = \frac{24}{5} = 4.8 $? No, 5 is not less than 4.8, so that's false.

Now, let's check $(2, 0)$ again. It works for both. Let's check $(1, -1)$:

### Step 1: First inequality
$ -1 < -\frac{2}{5}(1) + 6 $
$ -1 < -\frac{2}{5} + 6 $
$ -1 < \frac{28}{5} $ (true)

### Step 2: Second inequality
$ -1 < 5(1) - 6 $
$ -1 < -1 $ (false, since it's not less than, it's equal)

So, $(2, 0)$ works. Let's confirm with the graph. The dashed lines mean the inequalities are strict (<, not ≤). The line $ y = 5x - 6 $ passes through (1, -1) and (2, 4), (3, 9), etc. The line $ y = -\frac{2}{5}x + 6 $ passes through (0, 6), (5, 4), (10, 2), etc. The region below both lines would be where $ y $ is less than both. For $ x = 2 $, $ y < -\frac{4}{5} + 6 = 5.2 $ and $ y < 10 - 6 = 4 $. So the stricter condition is $ y < 4 $ when $ x = 2 $. So $ (2, 0) $ is in that region.

Another point: (3, 4) – wait, $ y = 4 $ for $ x = 3 $ in the second inequality: $ 4 < 5(3) - 6 = 9 $ (true), and in the first inequality: $ 4 < -\frac{6}{5} + 6 = 4.8 $ (true). Wait, but earlier I thought (3, 5) was false, which it is. So (3, 4) is also a solution. But let's check the graph. The dashed lines: the line with negative slope is $ y = -\frac{2}{5}x + 6 $ (dashed, since it's <), and the line with positive slope is $ y = 5x - 6 $ (dashed). The region below both would be to the right of the intersection point? Wait, no. Let's see the slopes: the first line has slope -2/5 (negative), the second has slope 5 (positive). So the first line is decreasing, the second is increasing. Their intersection is at (20/9, 46/9) ≈ (2.22, 5.11). So below both lines would be the region that is below the decreasing line and below the increasing line. Let's take a point to the right of the intersection, say (3, 0):

### Step 1: First inequality
$ 0 < -\frac{6}{5} + 6 = 4.8 $ (true)

### Step 2: Second inequality
$ 0 < 15 - 6 = 9 $ (true)

Wait, (3, 0) is also a solution. Wait, maybe I made a mistake earlier. Let's check (3, 0):

First inequality: $ 0 < -\frac{2}{5}(3) + 6 = -\frac{6}{5} + \frac{30}{5} = \frac{24}{5} = 4.8 $ (true)

Second inequality: $ 0 < 5(3) - 6 = 15 - 6 = 9 $ (true)

So (3, 0) is also a solution. Wait, maybe the answer is any point that satisfies both. But the problem says "identify which point", so maybe we need to check the options (but the user didn't provide options, so maybe in the original problem there were options). Wait, the user's image shows a graph with grid lines. Let's assume the options are, for example, (2, 0), (3, 5), (0, 6), (1, -1). Let's check (1, -1):

### Step 1: First inequality
$ -1 < -\frac{2}{5}(1) + 6 = 5.6 $ (true)

### Step 2: Second inequality
$ -1 < 5(1) - 6 = -1 $ (false, since it's not less than, it's equal)

(0, 6):

### Step 1: First inequality
$ 6 < 6 $ (false, since it's <, not ≤)

(3, 5):

### Step 1: First inequality
$ 5 < -\frac{6}{5} + 6 = 4.8 $ (false)

So the only point that works among these is (2, 0) or (3, 0), etc. But since the user didn't provide options, maybe in the original problem, the options included (2, 0) or similar. Assuming the options are, for example, A. (2, 0), B. (3, 5), C. (0, 6), D. (1, -1), then the correct answer is A. (2, 0).

# Answer:
A. (2, 0) (assuming this is one of the options; if options differ, adjust accordingly)

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QUESTION IMAGE

Question

Explanation:

Step 1: Test the first inequality \( y < -\frac{2}{5}x + 6 \)

Step 2: Test the second inequality \( y < 5x - 6 \)

Step 1: First inequality

Step 1: First inequality

Step 2: Second inequality

Step 1: First inequality

Step 2: Second inequality

Answer: