Question

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Answer 1

Answer: y < -4x+3

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We need to find the equation for the boundary line. It goes through the two points (1,-1) and (2,-5)

Find the slope of the line through (x1,y1) = (1,-1) and (x2,y2) = (2,-5)

m = (y2 - y1)/(x2 - x1)

m = (-5 - (-1))/(2 - 1)

m = (-5 + 1)/(2 - 1)

m = -4/1

m = -4 is the slope

Plug m = -4 and (x1,y1) = (1,-1) into the point slope formula. Solve for y.

y - y1 = m(x - x1)

y - (-1) = -4(x - 1)

y + 1 = -4(x - 1)

y + 1 = -4x + 4 ... distributing

y + 1 - 1 = -4x + 4 - 1 ... subtracting 1 from both sides

y = -4x + 3 is the equation of the boundary line

The blue shaded region tells us to consider all points below this boundary line as solutions. So we will use a "less than" sign to indicate this

We go from y = -4x+3 to y < -4x+3 which is the answer we want.

We do not have a line under the inequality sign. Simply because the boundary is a dashed line, so we exclude points on the boundary line as solutions. If we had
`y \le -4x+3`, then the boundary line would be solid and it would tell the reader "points on the boundary line are part of the solution set".

Answer 2

`y<-4x+3`

Step-by-Step Explanation:

We want to find the slope-intercept inequality for the graph shown.

First, we will need to determine the equation of the line. We are given two points: (1, -1) and (2, -5). Let’s use the two to determine the slope. The slope formula is given by:

`\displaystyle m=(y_2-y_1)/(x_2-x_1)`

Let (1, -1) be (x₁, y₁) and let (2, -5) be (x₂, y₂). Substitute appropriately:

`\displaystyle m=(-5-(-1))/(2-1)=(-4)/(1)=-4`

So, our slope is -4.

Now, we can use the point-slope form:

`y-y_1=m(x-x_1)`

Where m is the slope and (x₁, y₁) is a point.

So, let’s substitute -4 for m. For consistency, we will let (1, -1) be (x₁, y₁). Hence:

`\displaystyle y-(-1)=-4(x-1)`

Distribute:

`y+1=-4x+4`

Subtract 1 from both sides:

`y=-4x+3`

Finally, we can determine our sign.

Notice that our line is dotted. Therefore, we do not have “or equal to.”

Also, notice that the area shaded is below our line. Therefore, our y is less than our equation .

So, our symbol should be ”less than.”

Therefore, our equation is:

`y<-4x+3`