Answer:
We know that the shaded region is the area of intersection of the areas below/above the 2 lines
We will first find the equation of the increasing line using the slope-intercept formula
We can see that it intersects the y-plane at y = 4
Hence, it's y-intercept is 4
Now we will find the slope of the line:
Since the slope of a line is given by the formula slope = rise/run
since the rise of this line is equal to its run,
we can say that the slope of this line is 1
Equation of line 1:
ignoring the inequalities and using the slope-intercept formula, we get
y = x + 4
Now, we will talk about the inequality symbol
We know that the shaded area on the graph is the intersection of the inequalities
Since the shaded area is under the line, the area of this inequality is under the line, which means the area covers values which are lower than the value of the line
Hence, we can say that there will be a less-than (<) sign
We can also see that the line is not dotted, this means that the values of the inequality include the values on the line
Which means that instead of a less-than (<) sign, we will have a less-than or equal-to sign (≤)
Which means that the equation of line 1 will be:
y ≤ x + 4
Now on to the next line:
We can see that the y-intercept of this line is y = 1
Now, finding the slope:
Slope = Rise / Run
We notice that the rise = run = 1
Hence, slope of line 2 is also 1
but since this line is going down, its slope will be negative
Slope = -1
Equation of line 2:
Ignoring the inequality, the equation of this line according to the slope-intercept formula the equation of this line will be:
y = -x + 1 OR y = 1 - x
Now, we will talk about the inequality symbol:
We know that the area on the graph is the area of intersection of the areas of both the lines
Since the area of intersection is above this line,
we know that there will be a greater-than (>) sign
We also see that the line is not dotted, which means that the values on the line are also included in the area of the inequality
Which adds an equal-to to the sign
Hence, the symbol of inequality for line 2 is greater-than or equal-to (≥)
Therefore the equation of line 2:
y ≥ 1 - x
Finally, the 2 equations in this system of inequality are:
y ≥ 1 - x and y ≤ x + 4