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.
$y < -dfrac{2}{5}x + 6$
$y < 5x - 6$
graph of two dashed lines: one with y-intercept 6 and negative slope, one with y-intercept -6 and positive slope, intersecting on a grid with x from -10 to 10 and y from -7 to 10

Explanation:

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 some common test points (though the graph is given, we can analyze the regions).

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

This is a line with slope \( -\frac{2}{5} \) and y-intercept 6. The shading is below this line (since \( y < \) the line).

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

This is a line with slope 5 and y-intercept -6. The shading is below this line (since \( y < \) the line).

Step 3: Find the intersection of the two regions

The solution to the system is the region where both shadings overlap. Let's find the intersection point of the two lines (though we can test points).

First, find the intersection of \( y = -\frac{2}{5}x + 6 \) and \( y = 5x - 6 \):

Set \( -\frac{2}{5}x + 6 = 5x - 6 \)

Multiply both sides by 5 to eliminate the fraction:

\( -2x + 30 = 25x - 30 \)

\( 30 + 30 = 25x + 2x \)

\( 60 = 27x \)

\( x = \frac{60}{27} = \frac{20}{9} \approx 2.22 \)

Then \( y = 5(\frac{20}{9}) - 6 = \frac{100}{9} - \frac{54}{9} = \frac{46}{9} \approx 5.11 \)

Now, test a point in the overlapping region. Let's take \( (3, 4) \):

For \( y < -\frac{2}{5}x + 6 \): \( 4 < -\frac{6}{5} + 6 = -\frac{6}{5} + \frac{30}{5} = \frac{24}{5} = 4.8 \). True.

For \( y < 5x - 6 \): \( 4 < 15 - 6 = 9 \). True.

Another point: \( (2, 3) \)

For \( y < -\frac{2}{5}x + 6 \): \( 3 < -\frac{4}{5} + 6 = \frac{26}{5} = 5.2 \). True.

For \( y < 5x - 6 \): \( 3 < 10 - 6 = 4 \). True.

But let's check the graph. The line \( y = 5x - 6 \) crosses the x-axis at \( x = \frac{6}{5} = 1.2 \). The line \( y = -\frac{2}{5}x + 6 \) crosses the y-axis at 6.

The overlapping region is to the right of the intersection point (since the second line has a steeper positive slope) and below both lines.

A common point that works is \( (3, 4) \) (or other points in that region).

Final Answer

A point that satisfies both inequalities is, for example, \( \boldsymbol{(3, 4)} \) (or other valid points in the overlapping region).

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 some common test points (though the graph is given, we can analyze the regions).

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

This is a line with slope \( -\frac{2}{5} \) and y-intercept 6. The shading is below this line (since \( y < \) the line).

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

This is a line with slope 5 and y-intercept -6. The shading is below this line (since \( y < \) the line).

Step 3: Find the intersection of the two regions

The solution to the system is the region where both shadings overlap. Let's find the intersection point of the two lines (though we can test points).

First, find the intersection of \( y = -\frac{2}{5}x + 6 \) and \( y = 5x - 6 \):

Set \( -\frac{2}{5}x + 6 = 5x - 6 \)

Multiply both sides by 5 to eliminate the fraction:

\( -2x + 30 = 25x - 30 \)

\( 30 + 30 = 25x + 2x \)

\( 60 = 27x \)

\( x = \frac{60}{27} = \frac{20}{9} \approx 2.22 \)

Then \( y = 5(\frac{20}{9}) - 6 = \frac{100}{9} - \frac{54}{9} = \frac{46}{9} \approx 5.11 \)

Now, test a point in the overlapping region. Let's take \( (3, 4) \):

For \( y < -\frac{2}{5}x + 6 \): \( 4 < -\frac{6}{5} + 6 = -\frac{6}{5} + \frac{30}{5} = \frac{24}{5} = 4.8 \). True.

For \( y < 5x - 6 \): \( 4 < 15 - 6 = 9 \). True.

Another point: \( (2, 3) \)

For \( y < -\frac{2}{5}x + 6 \): \( 3 < -\frac{4}{5} + 6 = \frac{26}{5} = 5.2 \). True.

For \( y < 5x - 6 \): \( 3 < 10 - 6 = 4 \). True.

But let's check the graph. The line \( y = 5x - 6 \) crosses the x-axis at \( x = \frac{6}{5} = 1.2 \). The line \( y = -\frac{2}{5}x + 6 \) crosses the y-axis at 6.

The overlapping region is to the right of the intersection point (since the second line has a steeper positive slope) and below both lines.

A common point that works is \( (3, 4) \) (or other points in that region).

Final Answer

A point that satisfies both inequalities is, for example, \( \boldsymbol{(3, 4)} \) (or other valid points in the overlapping region).