EXPLANATION :
From the problem, we have the inequalities :
[tex]\begin{gathered} y>5x+1 \\ y\le-x+3 \end{gathered}[/tex]
Take the symbols as "=" sign.
We need two points to graph the inequalities.
y = 5x + 1
when x = 0, the value of y is :
y = 5(0) + 1
y = 1
when y = -4, the value of x is :
-4 = 5x + 1
-4 - 1 = 5x
-5 = 5x
x = -1
So we have the points (0, 1) and (-1, -4)
The type of boundary line depends on the inequality symbol.
Since the symbol is ">", the boundary line is a dashed or broken line.
Determine the region by testing the origin (0, 0)
If (0, 0) satisfies the inequality, then the region will pass thru the origin.
y > 5x + 1
0 > 5(0) + 1
0 > 1
False
Since the result is NOT true, the region will NOT pass thru the origin.
The graph will be :
Next is to graph the second inequality :
y = -x + 3
when x = 0, the value of y is :
y = -0 + 3
y = 3
when y = 0, the value of x is :
0 = -x + 3
x = 3
The points are (0, 3) and (3, 0)
The boundary line is a solid line since the symbol is "≤"
Determine the region of the second inequality by testing again the origin (0, 0)
y ≤ -x + 3
0 ≤ -0 + 3
0 ≤ 3
True
Since the result is true, the region will pass thru the origin.
The graph will be :
The solution is the overlapping region between the two inequalities.
