We want to graph the following system of inequalities.
1+x-y≥3x-2y-4 & (I) y<2x^2+1 & (II)
The solution set is the overlapping region of the solution sets of the inequalities. Let's graph each of them, one at a time.
Inequality (I)
To graph the inequality, we have to draw the boundary line. The equation of a boundary line is written by replacing the inequality symbol from the inequality with an equals sign.
Inequality:&
1+x-y ≥ 3x-2y-4
Boundary Line:& 1+x-y = 3x-2y-4
To draw this line, we will first rewrite the equation in slope-intercept form.
We will plot the y-intercept (0, -5), then use the slope m=2 to plot another point on the line. Connecting these points with a solid line will give us the boundary line of our inequality. Note that the boundary line is solid, not dashed, because the inequality is notstrict.
To decide which side of the boundary line to shade, we will substitute a test point that is not on the boundary line into the given inequality. If the substitution creates a true statement, we shade the region that includes the test point. Otherwise, we shade the opposite region. Let's use (0,0) as our test point.
Since the substitution of the test point created a true statement, we will shade the region that contains the point.
Inequality (II)
Similarly, we can write the boundary curve for Inequality (II) by replacing the less than sign with an equals sign. Note that this is a quadratic inequality. Let's identify a, b, and c.
y=2x^2+1 ⇔ y= 2x^2+ 0x+ 1
Knowing that a= 2, b= 0, and c= 1, we can find the vertex. To do so, we will need to think of y as a function of x, y=f(x).
Vertex of the Parabola: ( - b/2a,f(- b/2a) )
Let's substitute the values of a and b in the formula for the x-coordinate of the vertex.
The vertex is (0,1). With this information, we know that the axis of symmetry of the parabola is the vertical line x=0. Next, let's find two more points on the curve — one on each side of the axis of symmetry.
x
2x^2+1
y=2x^2+1
-1
2(-1)^2+1
3
1
2(1)^2+1
3
The points (-1,3) and (1,3) are on the parabola. Because we have a strict inequality, the boundary curve will be dashed. Let's plot the points and connect them with a smooth curve.
Now that we have the boundary curve, we need to determine which region to shade. To do so, we will use (0,0) as a test point.
