Looking at the shaded figure, imagine the boundary lines that make up this polygon. They pass through the given points and create a system of three linear inequalities.
We will be able to use the given points to write equations for the boundary lines. From there, we will use these equations to write the inequalities for the system.
The first boundary line passes through the points (-2,-3) and (4,0). Substituting these points in the , we can determine its m.
Since we know the slope of the boundary line and at least one point through which it passes, we can write its equation in . Let's use ( 4, 0).
Point-slope form:& y- y_2=m(x- x_2)
Boundary Line I:& y- 0=1/2(x- 4)
Now, to write the equation in let's isolate y.
Next, to determine the inequality sign, we first observe the boundary line to see if it is strict. In this case, the boundary line is solid, so the points that lie on it are included in the solution set.
We have identified a point on the graph that we know satisfies the inequality. By substituting this point into our inequality and simplifying, we can identify the correct inequality symbol.
y 1/2x-2
0 1/2( 0)-2
0 0-2
0 -2
We already know that the inequality is not strict, and substituting a point from the solution set created a statement that requires a greater than or equal to
symbol to be true. We can now complete this inequality.
Boundary Line I: y=1/2x-2
Inequality I: y≥1/2x-2
Inequality II
To write an equation for the second boundary line, we will first determine its m. We know that the line passes through the points ( 0, 3) and ( 4, 0), so we can use these in the .
m = y_2-y_1/x_2-x_1
m=0- 3/4- 0
m=-3/4
We know that the point ( 0, 3) is the y-intercept of the line, so with the slope m=- 34 and this point we can write an equation in .
Slope-Intercept Form: y = mx+b
Boundary line II: y = -3/4x+3
Next, to determine the inequality sign, we first observe the boundary line to see if it is strict. In this case the boundary line is solid, so the points that lie on it are included in the solution set.
We have identified a point on the graph that we know satisfies the inequality. By substituting this point into our inequality and simplifying, we can identify the correct inequality symbol.
y -3/4x+3
0 -3/4( 0)+3
0 0+3
0 3
We already know that the inequality is not strict, and substituting a point from the solution set created a statement that requires a less than or equal to
symbol to be true. We can now complete this inequality.
Boundary Line II: y=-3/4x+3
Inequality II: y≤-3/4x+3
Inequality III
To write an equation for the third boundary line, we will first determine its slope m. We know that the line passes through the points ( -2, -3) and ( 0, 3), so we can use these in the Slope Formula.
m = y_2-y_1/x_2-x_1
m=3-( -3)/0-( -2)
m=3+3/0+2
m=6/2
m=3
We know that the point (0,3) is the y-intercept of the line, so with the slope m=3 and this point we can write an equation in .
Slope-Intercept Form: y = mx+b
Boundary line III: y = 3x+3
To determine the inequality sign, we observe the boundary line to see if it is strict. In this case the boundary line is solid, so the points that lie on it are included in the solution set.
We've identified a point on the graph that we know satisfies the inequality. By substituting this point into our inequality and simplifying, we can identify the correct inequality symbol.
We already know that the inequality is not strict, and substituting a point from the solution set created a statement that requires a less than or equal to
symbol to be true. We can now complete this inequality.
Boundary Line III :& y=3x+9
Inequality III :& y≤ 3x+9
Writing the System
We combine all of the inequalities to have a completed system of inequalities for the given shaded figure.
y≥ 12x-2
y≤ - 34x+3
y≤ 3x+3
