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

Note: The function is below the graph, so it is less than.